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ELEMENTS   OF   ALGEBRA 


A  COURSE  FOR  GRAMMAR  SCHOOLS  AND  BEGINNERS 
IN  PUBLIC  AND  PRIVATE  SCHOOLS 


BY 


WILLIAM   J.   MILNE,   Ph.D.,    LL.D. 
Pbesidbiit  of  Nbw  York  State  Normal  College,  Albany,  N.Y. 


o»:o 


NEW   YORK:.  CINCINNATI:- CHICAGO 

AMERICAN   BOOK    COMPANY 


Copyright,  1894,  by 
AMEKICAN  BOOK  COMPANY. 


[All  rights  reserved.] 


MILNE'S   EL.    ALGc 


f>rinte&  b^ 

mniiiam  ITvison 

«ew  l!?orh,  xa.  S.  a. 


PREFACE. 


Algebra  has  not  always  proved  to  be  an  interesting 
subject  to  the  younger  classes  in  our  secondary  or  lower 
schools ;  indeed,  in  very  many  instances  it  has  been  greatly 
disliked  by  the  students  in  such  institutions.  Two  causes, 
chiefly,  have  conspired  to  produce  this  unfortunate  con- 
dition of  affairs,  —  one  the  unattractive  and  uninteresting 
method  of  presenting  the  subject ;  the  other,  the  difficulty 
of  the  examples  and  the  complexity  of  the  problems  pre- 
sented to  the  pupils  for  solution. 

It  is  believed  that  this  text-book  presents  the  elementary 
facts  of  the  science  in  such  a  manner  that  a  deep  interest 
will  be  awakened  in  algebraic  processes,  and  that  the 
examples  which  the  student  is  required  to  solve  are  quite 
within  the  scope  of  his  ability  to  master. 

The  author  has  in  several  instances  departed  from  the 
order  of  classification  commonly  followed  in  text-books  on 
algebra,  because  he  has  preferred  to  arouse  an  interest  in 
the  subject  rather  than  to  follow  an  order  which  is  scientific, 
but  which  does  not  introduce  the  student  to  the  attractive 
features  of  the  study  until  he  has  mastered  all  the  pro- 
cesses employed  with  most  forms  of  algebraic  quantities. 
And  yet  in  no  instance  have  erroneous  mathematical  ideas 


4  PREFACE. 

been  taught,  nor  has  correct  reasoning  ever  been  sacrificed 
for  the  purpose  of  exciting  such  interest. 

The  ideas  of  number  which  the  pupil  has  gained  in 
arithmetic  have  been  associated  with  those  involved  in 
algebra  in  such  a  way  that  no  difficulty  may  be  experienced 
in  passing  from  reasoning  upon  definite  numbers  to  reason- 
ing upon  general  numbers. 

The  treatment  of  equations  is  introduced  at  the  begin- 
ning, and  it  is  presented  throughout  the  book,  wherever  it 
is  possible  to  do  so  advantageously,  because,  since  ele- 
mentary algebra  treats  of  almost  nothing  except  the 
equation  and  general  numbers,  the  student  should  be  led 
to  a  comprehension  of  the  simpler  forms  of  the  equation 
as  soon  as  possible. 

The  method  of  presentation  exemplified  in  the  other 
books  of  the  series  has  been  followed  here,  because  it  is 
recognized  as  pedagogically  correct  and  because  it  has  met 
with  general  approbation. 

The  work  is  designed  to  present  the  merest  elements  of 
the  science,  and  yet  it  is  believed  that  the  method  of  pres- 
entation, the  illustration  and  application  of  mathematical 
principles,  and  the  knowledge  gained  from  the  solution  of 
problems,  will  familiarize  the  student  with  the  funda- 
mental principles  of  the  science  to  such  a  degree  that  easy 
and  rapid  progress  in  the  more  abstract  phases  of  the 
subject  will  be  secured  whenever  he  pursues  the  subject 

farther. 

WILLIAM  J.  MILNE. 
►State  Normal  College, 
Albany,  N.Y. 


CONTENTS. 


PAGE 

Algebraic  Processes 7 

Algebraic  Expressions 19 

Terms  in  Algebraic  Expressions 23 

Positive  and  Negative  Quantities 25 

Addition 20 

Subtraction 32 

Transposition  in  Equations 41 

Equations  and  Problems * 43 

Multiplication 47 

Special  Cases  in  Multiplication 54 

Simultaneous  P^quations 58 

Division 63 

General  Review  Exercises 70 

Factoring 73 

Equations  solved  by  Factoring 80 

Common  Divisors  or  Factors 85 

Common  Multiples 88 

Fractions       91 

Reduction  of  Fraction.s 92 

Clearing  Equations  of  Fractions      ...     * 101 

Addition  and  Subtraction  of  Fractions 107 

Multiplication  of  Fractions ,110 

Division  of  Fractions 114 

5 


6  CONTENTS. 

PAGE 

Equations 116 

Review  of  E(iuations .     .     ,     .  120 

Involution 131 

Evolution 134 

Quadratic  Equations 150 

Pure  Quadratic  Equations 150 

Affected  Quadratic  Equations 152 

Simultaneous  Quadratic  Equations 158 

General  Review 162 

Questions  for  Review 177 

Answers 184 


ELEMEJN^TS   OF   ALaEBRA. 


o><Ko 


ALGEBRAIC   PROCESSES. 

1.  Problem.  A  farmer  had  444  sheep  in  two  fields, 
one  of  which  contained  three  times  as  many  sheep  as  the 
other.     How  many  sheep  had  he  in  each  field  ? 

ARITHMETICAL    SOLUTIOX. 

A  certain  number  =  the  number  in  one  field. 
Then  3  times  that  number  =  the  number  in  the  other  field. 

And  4  times  that  number  =  the  number  in  both  fields. 

Therefore  4  times  that  number  =  444. 

The  number  =111,  the  number  in  one  field. 
3  times  111  =  333,  the  number  in  the  other  field. 

The  solution  given  above  may  be  abbreviated  by  using 
the  letter  n  for  the  expressions  a  certain  number  and  that 
yiumher.  In  algebra,  however,  it  is  customary  to  represent  a 
numbir  whose  value  is  to  be  found  by  x  or  by  some  other 
one  of  the  last  letters  of  the   alphabet. 

ALGEBRAIC    SOLUTION. 

Let  X  =  the  number  in  one  field. 

Then  3  a;  =  the  number  in  the  other  field. 

And  4  aj  =  the  number  in  both  fields. 

Therefore  4  x  =  444. 

X  =  111,  the  number  in  one  field. 
3ac  =  333,  the  number  in  the  other  field. 
7 


8  ELEMENTS   OF   ALGEBRA. 

2.  The  student  will  observe  that  algebra  is  very  much 
like  arithmetic,  but  it  differs  from  arithmetic  in  often  using 
letters  to  represent  numbers. 

Letters  used  to  represent  numbers  are  usually  called 
quantities,  though  it  is  also  proper  to  call  them  numbers. 

Thus,  X  in  the  preceding  algebraic  solution  is  called  a  quantity. 

3.  Letters  used  to  represent  numbers  or  quantities  whose 
values  are  to  be  found  are  called  Unknown  Numbers  or  Quan- 
tities. 

The  last  letters  of  the  alphabet,  as  x,  y,  z,  etc.,  are  com- 
monly employed  to  represent  unknown  numbers  or  quantities. 

Thus,  X  in  tlie  solution  already  given  is  called  an  unknown  quan- 
tity. 

4.  An  expression  of  equality  between  two  numbers  or 
quantities  is  called  an  Equation. 

Thus,  5  +  8  =  13,  and  3x  =  15  are  called  equations. 

5.  The  sign  =  is  used  to  indicate  equality. 

6.  The  sign  .-.is  used,  in  writing  solutions  of  problems, 
instead  of  the  word  therefore  or  hence.  It  is  called  the 
sign  of  Deduction. 

PROBI.EMS. 

7.  4I.  John  and  James  together  had  40  marbles,  but  John 
had  three  times  as  many  as  James.     How  many  had  each  ? 

2.  Two  boys  together  dug  60  bushels  of  potatoes.  If  one 
dug  twice  as  many  bushels  as  the  other,  how  many  did  each 
dig? 

3.  If  A  and  B  together  have  f  3000,  and  A  has  twice  as 
much  as  B,  how  much  has  each  of  them  ? 


ALGEBRAIC   PROCESSES.  9 

4.  A  number  added  to  five  times  itself  equals  30.  What 
is  the  number  ? 

^.  If  a  man  is  three  times  as  old  as  his  son,  and  the 
sum  of  their  ages  is  40  years,  how  old  is  the  father  ?  How 
old  is  the  son  ? 

6.  Divide  the  number  75  into  two  parts  such  that  one 
part  may  be  four  times  the  other. 

7.  A  man  bought  a  horse  and  carriage  for  $600,  and 
the  carriage  cost  him  twice  as  much  as  the  horse.  What 
did  each  cost  him  ? 

8.  A  certain  orchard  has  700  trees  in  it.  There  are 
twice  as  many  cherry  trees  as  pear  trees,  and  twice  as  many 
apple  trees  as  cherry  trees.  How  many  are  there  of  each 
kind? 

9.  A,  B,  and  C  were  talking  of  their  ages.  A  said,  "  I 
am  twice  as  old  as  B  " ;  B  said,  ^'  I  am  twice  as  old  as  C." 
The  sum  of  their  ages  was  140  years.     How  old  was  each  ? 

Suggestion.  Discover  from  the  problem  the  number  which,  if 
known,  will  enable  you  to  find  all  the  required  numbers,  and  let  x 
stand  for  that  number.     In  this  problem  x  will  stand  for  C's  age. 

10.  The  greater  of  two  luimbers  is  six  times  the  less,  and 
their  sum  is  49.     What  are  the  numbers  ?  \^ 

11.  After  taking  five  times  a, number  from  fifteen  times 
the  same  number,  and  adding  to  the  remainder  six  times 
the  number,  the  result  was  8  more  than  72.  What  was  the 
number  ? 

12.  Mary  has  five  times  as  many  books  as  Hannah, 
Hannah  has  two  times  as  many  as  Jane,  and  together  they 
have  39  books.     How  many  has  each  ? 


lb  ELEMENTS  OF   ALGEBRA. 

13.  I  paid  five  times  as  much  for  pans  as  for  ink,  and 
three  times  as  much  for  paper  as  for  both  of  these.  How 
much  did  I  pay  for  each,  if  they  all  cost  me  96  cents  ? 

14.  A  man  paid  twice  as  much  for  a  pair  of  trousers  and 
seven  times  as  much  for  an  overcoat  as  for  a  hat.  How 
much  did  each  cost  him,  if  they  all  cost  $60? 

15.  A  man  divided  $50  among  four  boys.  To  the  second 
he  gave  twice,  to  the  third  three  times,  and  to  the  fourth 
four  times,  as  much  as  to  the  first.  How  much  did  he  give 
to  each  ?     f 

16.  A  farmer  has  four  flocks  of  sheep.  The  second  is  twice 
as  large  as  the  first,  the  third  three  times  as  large,  and  the 
fourth  as  large  as  the  first  and  third  together.  If  he  has 
200  sheep  in  all,  how  many  are  there  in  each  flock  ? 

17.  If  one  number  is  six  times  another  number,  and  a 
third  number  is  10  times  that  number,  and  the  sum  of  the 
three  is  119,  what  are  the  numbers  ? 

18.  A  man  failed  in  business,  owing  A  ten  times  as  much 
as  B,  C  four  times  as  much  as  B,  and  D  eight  times  as  much 
as  the  difference  between  what  he  owed  A  and  C.  The 
entire  amount  he  owed  these  men  was  $  6300.  How  much 
did  he  owe  each  ? 

19.  If  a  gold  watch  is  worth  ten  times  as  much  as  a  silver 
watch,  and  both  together  are  worth  f  132,  how  much  is 
each  watch  worth  ? 

20.  The  whole  number  of  votes  cast  for  A  and  B  at  a 
certain  election  was  450,  but  four  times  as  many  votes  were 
cast  for  A  as  for  B.     How  many  were  cast  for  ea^h  ? 

21.  A  man  paid  $175  for  a  horse,  a  cow,  and  a  wagon. 
He  paid  twice  as  much  for  the  horse  as  for  the  wagon,  and 


ALGEBRAIC   PROCESSES.  11 

twice  as  much  for  the  wagon  as  for  the  cow.     How  much 
did  he  pay  for  each  ? 

22.  A,  B,  and  C  form  a  partnership  with  a  capital  of 
$  14,000.  B  and  C  each  furnish  three  times  as  much  as  A. 
How  much  capital  does  each  furnish  ? 

23.  A  and  B  performed  a  certain  piece  of  work  for  which 
B  received  three  times  as  much  as  A,  and  A  received  f  12 
less  than  B.     How  much  did  each  receive  ? 

Solution.     Let  x  =  the  amount  A  received. 
Then  Sx  =  the  amount  B  received. 

2x  =  the  amount  B  received  more  than  A. 

X  =  $  6,  the  amount  A  received. 
3ic  =  $18,  the  amount  B  received. 

24.  A  farmer  raised  five  times  as  many  acres  of  wheat 
as  of  oats,  and  live  times  as  many  acres  of  oats  as  of 
potatoes.  If  186  acres  were  cultivated,  how  many  acres  were 
there  of  each  ? 

25.  A  man's  capital  doubled  for  three  successive  years. 
If  he  then  had  f  9600,  how  much  had  he  at  first  ?  /OxT^ 

Suggestion.     Let  x  =  his  original  capital. 

Then  2  x  =  his  capital  at  end  of  first  year. 

26.  Sarah  had  seven  times  as  many  chickens  as  ducks,  and 
twice  as  many  ducks  as  hens.  If  there  were  in  all  119  fowls, 
how  many  were  there  of  each  kind  ? 

27.  A  certain  number  can  be  separated  into  two  factors, 
one  of  which  is  three  times  the  other,  and  whose  sum  is  36. 
What  are  the  two  factors  ? 

28.  One  cask  liolds  four  times  as  much  as  another,  and 
the  larger  contains  27  gallons  more  than  the  smaller.  How 
many  gallons  are  there  in  each  cask  ? 


12  ELEMENTS   OF   ALGEBRA. 

29.  The  population  of  one  town  is  eight  times  the  popu- 
hition  of  another,  and  the  larger  one  contains  4900  more 
inhabitants  than  the  smaller.  What  is  the  population  of 
each  town  ? 

30.  The  difference  between  a  certain  number  and  that 
number  multiplied  by  18,  is  119.     What  is  the  number  ? 

31.  A,  B,  and  C  work  the  same  number  of  days.  A 
receives  75  cents  per  day,  B  receives  90  cents,  and  C 
receives  95  cents.  All  together  receive  $  13.  How  many 
days  does  each  work  ? 

32.  A  father's  age  is  five  times  the  age  of  his  son,  and  the 
difference  between  their  ages  is  44  years.  What  is  the  age 
of  each? 

33.  What  number  added  to  .V  of  itself  equals  12  ? 


Solution. 

Let 

X  =  the  number. 

Then 

x^\x=Vl 

Or 

lx  =  \2 
.'.  ix  =  4 

And 

x  =  S 

Hence  the  number 

is 

8 

'  34.    If  a  certain  number  is  added  to  i  of  itself,  the  sum 
is  20.     What  is  the  number  ? 

35.  If  to  six  times  a  certain  number  i  of  itself  is  added, 
the  sum  is  19.     What  is  the  number  ? 

36.  A  farmer  had  f  as  many  horses  as  cows,  and 
altogether  he  had  10  cows  and  horses.  How  many  of  each 
had  he  ? 

37.  One  of  my  pear  trees  bore  f  as  many  bushels  of 
pears  as  the  other,  and  both  together  bore  18  bushels.  How 
many  did  each  tree  bear  ? 


ALGEBRAIC   PROCESSf:S.  13 

38.  There  is  a  certain  number,  ^  of  which  exceeds  ^  of  it 
by  2.     What  is  the  number  ? 

39.  One  half  of  a  certain  number  exceeds  i  of  it  by  3. 
What  is  the  number  ? 

40.  Divide  45  into  two  parts  such  that  one  may  be  f  of 
the  other. 

41.  One  half  and  one  third  of  a  certain  number  added 
together  make  10.     What  is  the  number  ? 

42.  Three  fourths  of  a  number  added  to  two  fifths  of 
it  makes  23.     What  is  the  number  ? 

43.  If  Charles  had  five  times  as  many  marbles  as  he  now 
has,  he  would  have  as  many  as  John  and  William  together, 
the  former  of  whom  has  30,  and  the  latter  15.  How  many 
marbles  has  Charles  ? 

44.  If  a  fourth  of  a  number  is  subtracted  from  \  the 
number,  the  result  is  3.     What  is  the  number  ? 

45.  A  man  had  260  sheep  in  three  fields.  In  the  second 
he  had  twice  as  many  as  in  the  first,  and  in  the  third  \ 

__of  the  number  in  the  first.    How  many  had  he  in  each  field  ? 

46.  The  less  of  two  numbers  is  4-  of  the  greater,  and 
their  sum  is  14  less  than  50.     What  are  the  numbers  ? 

47.  Divide  120  into  three  such  parts  that  the  first  part 
is  ^  of  the  second,  and  ^  of  the  third. 

SuGGESTiox.     Let  X  =  the  first  part. 

Then  2  x  =  the  second  part. 

48.  A,  B,  and  C  together  have  $900.  A  has  twice  as 
much  as  B,  and  C  has  twice  as  much  as  A  and  B  together. 
How  much  has  each  ? 

49.  What  number  is  that  to  which  if  you  add  its  half 
and  take  away  its  third,  the  remainder  will  be  147  ? 


14  ELEMENTS  OF   ALGEBRA. 

50.  James  has  ^  as  many  marbles  as  Henry,  and  Henry 
has  16  more  than  James.     How  many  marbles  has  each  ? 

51.  Julia  has  ^  as  much  money  as  May;  Hattie  has  ^ 
as  much  as  Julia ;  all  together  have  18  cents.  How  many 
cents  has  each  ? 

52.  A  certain  number  plus  i  of  itself,  plus  \  of  itself 
equals  46.     What  is  the  number  ? 

^  63.  In  Mr.  Green's  library  there  were  1900  volumes.  The 
number  of  books  of  fiction  was  |  the  number  on  science ; 
the  number  on  science  was  twice  the  number  on  history. 
How  many  volumes  were  there  of  each  ? 

54.    What  number  increased  by  ^V  ^^  itself  equals  133  ? 

-  "^  55.    B  has  f  as  much  money  as  A  ;  C  has  ^  as  much  as  B  ; 
altogether  they  have  $  27.     How  much  has  each  ? 

56.  Two  fifths  of  a  number  plus  ^  of  the  number  equals 
36.     What  is  the  number  ? 

57.  George  sold  a  certain  number  of  oranges  at  2  cents 
apiece  ;  Walter  sold  ^  as  many  at  4  cents  apiece.  If  both 
together  received  90  cents,  how  many  oranges  did  each  sell  ? 

Solution.     Let  x  =  the  number  George  sold. 
Then  i  ^  =  the  number  Walter  sold. 

2x  =  the  number  of  cents  George  received. 

^x  =  the  number  of  cents  Walter  received. 
I  a:  +  2  X  =  the  number  of  cents  both  received. 

.-.  -v-»  =  90. 

ix  =  9. 

X  =  27,  the  number  of  oranges  George  sold. 
1  a:  =  9,  the  number  of  oranges  Walter  sold. 


,J 


ALGEBRAIC   PROCESSES.  15 

58.  Robert  bought  a  certain  number  of  apples,  and  his 
mother  gave  him  4  as  many  as  he  bought.  He  then  had  18. 
How  many  apples  did  he  buy  ? 

69.  A  man  gave  $  50  to  three  poor  families.  To  the  first 
he  gave  ^  as  much  as  to  the  second,  to  the  third  seven  times 
as  much  as  to  the  first.     How  much  did  he  give  to  each  ? 

60.  Three  newsboys  sold  100  papers.  The  first  and  the 
second  sold  the  same  number,  while  the  third  sold  |  as 
many  as  the  other  two.     How  many  did  each  sell  ? 

61.  A  and  B  start  in  business  with  the  same  amount 
of  money.  A  gains  ^  as  much  as  he  had,  while  B  loses  ^of 
his  capital ;  they  then  have  together  $  1150.  What  was  the 
original  capital  of  each  ? 

62.  What  number  increased  by  i  of  |  of  itself  equals  52  ? 

i  63.  The  length  of  a  field  is  l^  times  its  breadth,  and  the 
entire  distance  around  the  field  is  60  rods.  What  are  the 
length  and  the  breadth  of  the  field  ? 

Solution.    Let  x  =  the  number  of  rods  in  the  breadth. 
Then  1 J  ic  =  the  number  of  rods  in  the  length. 

And  2^x  =  half  the  distance  around  the  field. 

.*.  6x  =  60  rods,  the  distance  around  the  field, 
ic  =  12  rods,  the  breadth  of  the  field, 
li  5c  =  18  rods,  the  length  of  the  field. 

64.  What  is  the  number  whose  double  and  half  added 
together  give  35  ? 

65 .  The  sum  of  three  numbers  is  34.  The  first  multiplied 
by  4  produces  the  second,  and  i  of  the  first  taken  from  the 
second  gives  the  third.     What  are  the  numbers  ? 

66.  Find  two  factors  of  100,  one  of  which  is  four  times 
the  other,  and  whose  sum  is  i  of  100. 


16  ELEMENTS   OF   ALGEBRA. 

67.  A  father  gave  to  his  first  son  three  times  as  much  as 
to  the  second,  to  the  third  ^  as  much  as  to  the  first  and 
second ;  in  all  he  gave  f  600.     How  much  did  he  give  to 

•  each  ? 

68.  Four  men  are  taxed  in  the  aggregate  f  500  as  follows  : 
for  every  dollar  that  the  first  pays  the  second  pays  1  dollar, 
the  third  pays  ^  of  a  dollar,  and  the  fourth  pays  11  dollars. 
How  much  is  each  man's  tax  ? 

69.  Emma  sold  10  quarts  of  berries  at  a  certain  price  per 
quart.  Anna  sold  14  quarts  at  twice  that  price  per  quart. 
Together  they  received  $1.90.  What  price  per  quart  did 
each  receive  ? 

70.  Two  boys  walked  toward  each  other  from  two  towns 
18  miles  apart.  If  one  boy  walked  ^  as  far  as  the  other, 
how  many  miles  did  each  walk  ? 

71.  What  number  is  that  whose  fourth  part  exceeds  its 
fifth  part  by  3  ? 

72.  A  man  paid  a  certain  sum  for  a  house  and  lot,  and 
sold  them  at  a  gain  equal  to  ^  the  cost.  If  the  selling  price 
>vas  f  3000,  what  was  the  cost  ? 


Solution. 

Let 

X  =  the  cost  of  the  house  ai 

id  lot. 

Then 

lx=  the  gain. 

And 

• 

l^x  =  the  sellmg  price. 
.  |x  =  3000. 
Jx  =  1000. 

X  =  2000,  the  cost  of  the  house  and  lot. 

73.  A  man  paid  a  debt  of  $7500  in  4  months,  paying 
each  month  twice  as  much  as  the  month  before.  How 
much  did  he  pay  the  first  month  ? 


ALGEBRAIC   PROCESSES.  17 

74.  A  man  traveled  324  miles;  he  went  three  times  as 
far  by  steamboat  as  by  stage,  and  eight  times  as  far  by 
railroad  as  by  steamboat.  How  many  miles  did  he  travel 
by  each  conveyance  ? 

75.  Harry  solved  8  problems  more  than  Edward,  and 
Edward  solved  ^  as  many  as  Harry.  How  many  did  each 
solve  ? 

76.  A  man  paid  $150  for  a  wagon,  which  was  50  per 
cent  or  one  half  more  than  the  original  cost.  What  was 
the  original  cost  of  the  wagon  ? 

77.  A.  man  bought  a  lot  for  a  certain  sum ;  he  built  upon 
it  a  house  costing  twice  as  much  as  the  lot ;  he  furnished 
the  house  at  an  expense  equal  to  i  the  price  of  the  lot. 
The  man,  who  had  only  f  6000  in  cash,  found  that  he  could 
pay  for  all  his  expenditures  except  furnishing  the  house. 
What  was  the  amount  of  each  item  ? 

78.  In  an  orchard  there  are  210  trees,  arranged  in  rows. 
There  are  5  rows  with  a  certain  number  of  trees  in  a  row, 
and  8  rows  with  twice  that  number  of  trees  in  a  row.  How 
many  trees  are  there  in  the  different  rows  ? 

79.  A  man  earned  daily  for  4  days  three  times  as  much 
as  he  paid  for  his  board.  After  paying  his  board  for  4 
days,  he  had  $  8  left.  How  much  did  he  receive  per  day, 
and  how  much  did  he  pay  for  his  board  ? 

80.  A  man  on  being  asked  how  old  he  was  said  that  y^^^ 
of  his  age  added  to  twice  his  age  made  84  years.  How  old 
was  he  ? 

81.  A  man  borrowed  as  much  money  as  he  had,  and  then 
spent  i  of  the  whole  sum.  If  he  had  f  4  left,  how  much 
had  he  at  first  ? 

MiLNE's    EL.    OF    ALG.  — 2. 


18  ELEMENTS  OF   ALGEBRA. 

82.  A  man  had  twice  as  many  5-cent  pieces  as  dimes, 
and  his  money  amounted  to  200  cents.  How  many  pieces 
had  he  of  each  kind  ? 

Solution.     Let  x  —  the  number  of  dimes. 

Then  "Ix  —  the  number  of  5-cent  pieces. 

And  10  X  =  the  number  of  cents  in  the  dime  pieces. 

10  X  =  the  number  of  cents  in  5-cent  pieces. 
.-.  20x==200. 

X  —  10,  the  number  of  dimes. 
2  X  =  20,  the  number  of  5-cent  pieces. 

83.  A  sum  of  money  amounting  to  f  5  is  composed  of 
half  and  quarter  dollars.  The  number  of  half  dollars  is 
double  the  number  of  quarter  dollars.  How  many  pieces 
are  there  of  each  kind  ? 

84.  A  man  worked  10  days  for  a  certain  sum  per  day ; 
his  wife  also  worked  at  the  same  rate,  and  his  son  at  \  the 
rate  of  his  father.  They  received  in  all  $27.50.  What 
were  the  daily  wages  of  each  ? 

85.  In  a  certain  field  the  length  is  twice  the  breadth, 
and  the  distance  around  the  field  is  90  rods.  What  are  the 
length  and  the  breadth  of  the  field  ? 

86.  A  tree  45  feet  high  was  broken  off  so  that  the  part 
left  standing  was  twice  the  part  broken  off.  What  was  the 
length  of  each  part? 

87.  Divide  $800  among  three  men  so  that  the  first  and 
third  shall  together  have  three  times  as  much  as  the  second, 
and  the  first  shall  have  double  what  the  third  has. 

88.  The  income  from  a  very  successful  business  quadru- 
pled every  year  for  three  years.  If  the  entire  income  for 
the  three  years  was  $42000,  what  was  the  income  for  each 
year? 


ALGEBRAIC   EXPRESSIONS.  19 

ALGEBRAIC  EXPRESSIONS. 

8.  Quantities  connected  by  algebraic  signs  are  called 
Algebraic  Expressions. 

Thus,  a  +  b  and  2  a  —  3  he  are  algebraic  expressions. 

9.  The  signs  employed  in  arithmetic  are  generally  used 
for  the  same  purposes  in  algebra. 

10.  What  do  the  following  expressions  indicate  ? 

1.  2  4-5;  17-6;  16 -- 4  ;  15  x  3. 

2 .  a  +  6  ;  c  —  f/ ;  c  -^  d ;  d  x  a. 

3.  ff  +  6  -h  c  ;  «  —  &  -f  c ;  a  —  b  —  c;  a  X  b  X  c 

4.  a  —  b  —  c-\-cl:  a  -\-b  —  d  —e;  a  x  b  x  c  X  d. 

M lilt ipUcat ion  is  also  indicated  by  writing  letters,  or  a  letter  and  a 
figure,  side  by  side  without  any  sign  between  them,  or  with  a  dot  be- 
tween them.  Thus,  a  x  b  x  c  may  be  written  abc  or  a-b-  c.  So,  also, 
2  X  X  X  y  X  z  may  be  written  2 xyz  or  2  -x-  y  -  z. 

What  do  the  following  expressions  indicate  ? 

5.  4a6;  5a6;  3a  4-2&;  4ic -f  3.V. 

6.  4  xyz ;  6  abc ;  6ax  -\-3ay;  5cd  —  3d. 

7.  3xy  —  4:yz  -\-2z',  'i xyz  —  2 axy  —  3  by. 

8.  5 xyz  -\-  4:xy  —  3y;   2 a&c  —  abd  -j-  3 rf. 

11.  The  product  arising  from  using  a  quantity  a  certain 
number  of  times  as  a  factor  is  called  a  Power  of  that 
quantity. 

Thus,  4  is  a  power  of  2  ;  64  is  a  power  of  4  and  also  of  8. 

Powers  are  indicated  by  a  small  figure  or  letter,  called  an 
Exponent,  written  a  little  above  and  at  the  right  of  the 
quantity,  showing  the  number  of  times  the  quantity  is  to  be 
used  as  a  factor. 

Thus,  a^  shows  that  a  is  to  be  used  as  a  factor  ^'i?e  times,  or  that  it 
is  equal  toaxaxaxrtxn:. 


20  j:lements  of  algebra. 

Powers  are  named  from  the  number  of  times  a  quantity 
is  used  as  a  factor. 

Thus,  a'  is  read  the  seventh  power  of  a  or  a  seventh. 

The  second  power  is  also  called  the  square,  and  the  third 
power  the  cube  of  a  quantity. 

12.  Eead  the  following  expressions  and  state  what  they 
indicate. 

1.  a^',  a^\  x^]  ^y^'i  ^'Vj  ^y?  ^V- 

2.  d^^h-,  a-'+&';  o?-\-h^\  a^-h\ 

3.  d^h  -\-  alP- ;  c^x  —  Jfy'^ ;  aV  +  h^x^ ;  d^x^y  —  Ifx^y^. 

4.  a-  4-  ^2  _  ^2  .     ^2^2  ^_  ^2^2  _|_  ^2^2  .     ^^^3  ^  ^^^2^2  _  ^3^ 

13.  A  figure  or  letter  placed  before  a  quantity  to  show 
how  many  times  it  is  taken  additively  is  called  a  Coefficient. 

Thus,  in  the  expression  5  ic,  5  indicates  that  x  is  to  be  taken  five 
times,  or  that  it  is  equal  to  x  -\-  x  -\-  x  -\-  x  -\-  x. 

In  the  expression  4  6c,  4  may  be  regarded  as  the  coefficient  of  ftc,  or 
4  h  may  be  regarded  as  the  coefficient  of  c. 

When  no  coefficient  is  \vritten,  it  is  manifestly  1. 

14.  What  do  the  following  expressions  indicate  ? 

1.   bd'x'^. 

Interpretation.  The  expression  indicates  five  times  the  product 
of  a  used  twice  as  a  factor,  multiplied  by  x  used  three  times  as  a  factor, 
multiplied  by  z  used  five  times  as  a  factor. 

It  is  usually  read^ve  a  square^  x  cube,  z  fifth. 

2.  ^x'y^  5.    2x2  +  2/'.  8.    :^x'-\-2y-3z. 

3.  4.^y\  6.    3x-4.y\  9.    2d'x' -'dxy -^2%^ 

4.  6axy^.  7.    4:xy  —  3f.  10.    5a^y+3xY-^^cz'. 

15.  When  several  quantities  are  inclosed  in  parentheses, 
(),  they  are  to  be  subjected  to  the  same  process. 


ALGEBRAIC   EXPRESSIONS.  21 

Thus,  2  X  (x  +  ^)  indicates  two  times  the  sum  of  x  and  y ; 
3 (a  +  6  —  c)  indicates  three  times  the  remainder  when  c  has  been 
subtracted  from  the  sum  of  a  and  h. 

16.    1.    Interpret  and  read  the  expression  a  x  (6  +  c)  or 

Interpretation.  The  expression  means  that  the  sum  of  h  and  c 
is  to  be  multiplied  by  a.  It  is  usually  read  a  times  the  quantity  h 
plus  c. 

Interpret  and  read  the  following : 

2.  x(y-\-z).  7.    a-{-4:d(ax  —  cy). 

3.  5(a-c).  8.    3x  +  4.{2y-3z), 

4.  3d(c-y).  9.   2(3x-5y)-h{6x-3y). 

5.  4aj(c-t-2d).  10.    3a(x-{-c)  — oy{z-{-d), 

6.  3ac(2x  —  3y^).  11.    5aa.'(a  —  6)  — ;^c^(c  +  d). 

Write  the  following  in  algebraic  expressions  : 

1.  The  sum  of  five  times  a  and  three  times  the  square  of  x. 
Solution.     5  «  +  3  x^, 

2.  Three  times  b,  diminished  by  5  times  a  raised  to  the 
fourth  power. 

3.  The  product  of  a,  b,  and  a  —  c. 

4.  Seven  times  the  product  of  x  times  y,  increased  by 
three  times  the  cube  of  z. 

5.  Three  times  x,  diminished  by  five  times  the  sum  of  a, 
b,  and  c. 

6.  Six  times  the  square  of  m,  increased  by  the  product  of 
m  and  tl 

7.  The  product  of  a  used  five  times  as  a  factor,  multiplied 
by  the  sum  of  b  and  c. 

8.  Twelve  times  the  square  of  a,  diminished  by  five  times 
the  cube  of  b. 


22  ELEMENTS  OF   ALGEBRA. 

9.    Eight  times  the  product  of  a  and  b,  diminished  by  four 
times  the  fourth  power  of  c,  or  c  used  four  times  as  a  factor. 

10.  Six  times  the  product  of  the  second  power  of  a  mul- 
tiplied by  71,  increased  by  five  times  the  product  of  a  times 
the  second  power  of  7i. 

11.  Four  times  the  product  of  x  by  the  second  power  of 
y,  multiplied  by  the  cube  of  z,  diminished  by  a  times  the 
fourth  power  of  c. 

12.  The  fourth  power  of  a  plus  the  cube  of  b,  plus  three 
times  the  product  of  the  square  of  a  by  the  square  of  6, 
diminished  by  the  cube  of  d. 

17.  When  a  =  1,  b  =  2,  c  =  3,  d  =  4,  e  =  5,  find  the  nu- 
merical value  of  each  of  the  following  expressions  by  using 
the  number  for  the  letter  which  represents  it. 

Thus,  2a-[-36-c2-f(?  =  2-|-6-9H-4  =  3. 

1.  3 abed.  13.  12 be  +  e(c  —  a). 

2.  oabcd'.  14.  3crd  —  5abc. 

3.  2a'b'c\  15.  9a-4.b-\-2cd. 

4.  3  abode.  16.  3  abd  —  ^  e -\- cd. 

5.  3a-\-:)b.  17.  d'y-c- —  d- ■\- e\ 

6.  4a-fo6.  18.  {a-\-b)-\-e\d  —  c). 

7.  2e—3c-\-d.  19.  b{d  -  a -^  c)  -  bd. 

8.  4  ab-cd'.  20.  abc  +  ade  —  d{e  —  c). 

9.  10  d^b  — 3  ab-.  21.  b'C-d'- —  ab -^  cd. 

10.  3d  —  2b'-'{-e.  22.    b{d -^  a)-\-e{b -\-  c). 

11.  5cd  —  5bc  —  oab.         23.    c^-f- 4a&c  4- d(e  —  d). 

12.  Sa'b'^-\-c^-de.  24.    {a-\-b)-\-3{c^d)  ~{-2(?-d). 


ALGEBRAIC   EXPRESSIONS.  23 


TERMS  IN  ALGEBRAIC  EXPRESSIONS. 

18.  An  algebraic  expressiou  whose  parts  are  not  sepa- 
rated by  the  signs  +  or  —  is  called  a  Term. 

Thus,  5  X,  3  by,  4  x^y'^z^^  a%hl^e  are  terms. 

In  the  expression  3  x  +  2  ay  +  3  cd  —  ^  there  are  four  terms  ;  in  the 
expression  xy'^  —  x^y'^  there  are  but  two  terms. 

19.  A  term  which  has  the  sign  -h  before  it  is  called  a 
Positive  Term.  When  the  first  term  of  an  expression  is 
positive,  the  sign  +  is  usually  omitted  before  it. 

Thus  in  the  expression  2a  —  36  +  4c  —  5(^+3  6  the  first,  third, 
and  fifth  terms  are  positive. 

20.  A  term  which  has  the  sign  —  before  it  is  called  a 
Negative  Term. 

Thus  in  the  expression  2a  —  35  +  4c  —  5  d—Z  e,  the  second,  fourth, 
and  fifth  terms  are  negative. 

21.  Terms  which  contain  the  same  letters  with  the  same 
exponents  are  called  Similar  Terms. 

The  coefficients  or  signs  need  not  be  alike,  however. 

Thus,  5x3  and  —  Ix^  are  similar  terms  ;  also  3(x  +  a)^  and  b{x+  a)^. 
Expressions  like  hx^  and  cx^  may  be  considered  similar  terms  by 
regarding  h  and  c  as  coefficients. 

22.  Terms  which  contain  different  letters,  or  the  same 
letters  with  different  exponents,  are  called  Dissimilar  Terms. 

Thus,  Zxy  and  Zpq  are  dissimilar  terms  ;  also  5  x^y  and  5  xy^. 

23.  An  algebraic  expression  consisting  of  but  one  term 
is  called  a  Monomial. 

Thus,  a&,  Zaxyz,  bx'^y'^z  are  monomials. 

24.  An  algebraic  expression  consisting  of  more  than  one 
term  is  called  a  Polynomial. 

Thus,  a  4-  6  +  c  and  3  x  +  4  ?/  are  polynomials. 


24  ELEMENTS  OF   ALGEBRA. 

0 

25.  A  polynomial  of  two  terms  is  called  a  Binomial. 
Thus,  8  X  +  2  ?/,  a  —  6,  and  4  a?)  —  3  cd  are  binomials. 

26.  A  polynomial  of  three  terms  is  called  a  Trinomial. 
Thus,  a  +  6  +  c  and  2x  —  2y  -\-  bz  are  trinomials. 

27.  Select  from  the  following  : 

1st,  the  positive  terms.  5th,  the  monomials. 

2d,  the  negative  terms.  6th,  the  binomials. 

3d,  the  similar  terms.  7th,  the  trinomials. 

4th,  the  dissimilar  terms.  8th,  the  polynomials. 

1.  2air,  3aj2?/,  2a  +  3&,  6a;V,  oaa-^  3a;-f-2&. 

2.  3  x^  +  2 1/^,  5  a  —  3  6  +  c,  —  4  ax,  Q  a  —  c  -{-  d,  a^  -f-  61 

3.  a  +  6  4-  2  c  +  3  d,  aaj  —  6a;  —  3 c  -f  2  d,  ax-\-  bx. 

4.  aV  +  2 ax,  3a  -\-b,  2a  —  c  -\-  d,  3 aV  —  4 ax. 

Write  the  following : 

5.  Four  positive  terms;  three  negative  terms. 

6.  An   expression   containing   three   positive   and   two 
negative  terms. 

7.  Three  similar  terms;  four  dissimilar  terms. 

'    8.    An  expression  containing  four  positive  similar  terms, 
and  one  containing  four  positive  dissimilar  terms. 

9.    An  expression  containing  five  similar  terms,  three  of 
which  are  positive  and  two  negative. 

10.  An  expression  containing  five  dissimilar  terms,  four 
of  which  are  negative  and  one  positive. 

11.  Six  positive  monomials;  three  positive  similar  mono- 
mials ;  four  negative  monomials ;  three  negative  similar 
monomials. 

12.  Four  polynomials ;  four  binomials  ;  four  trinomials. 


ALGEBRAIC    EXPRESSIONS.  25 

POSITIVE  AND  NEGATIVE  QUANTITIES. 

28.  In  arithmetic  the  signs  -|-  and  —  are  used  to  indicate 
operations  to  be  performed,  but  in  algebra  they  are  used 
also  as  Signs  of  Opposition.  > 

Thus  if  gains  are  considered  j:)osi^i>e  quantities,  losses  will  be  nega- 
tive; if  distances  north  from  a  given  point  are  considered  positive^ 
distances  sonth  will  be  considered  negative^  etc. 

1.  Place  appropriately  before  each  of  the  following  quan- 
tities the  sign  -|-  or  the  sign  —  : 

Mr.  A  gains  $40  and  loses  $20.  Mr.  S  earns  $25  and 
spends  $  15.  A  ship  sails  40  miles  north  from  a  given  merid- 
ian, and  then  20  miles  south.  A  thermometer  indicates  on 
Monday  15°  above  zero,  and  on  Wednesday  10°  below  zero. 

2.  A  man  deposits  in  a  bank  $50,  and  then  draws  out 
$20.    Indicate  the  transaction  by  using  the  signs  -f  and  — . 

3.  Mr.  H  buys  25  horses  and  sells  10  of  them;  he  then 
buys  15  more  and  sells  8  of  them.  Indicate  the  transac- 
tions by  proper  signs. 

4.  A  vessel  sailed  40  miles  east,  and  was  then  driven  by 
adverse  winds  30  miles  west.  Indicate  the  directions  by 
proper  signs. 

5.  How  much  worse  off  is  a  man  who  is  $50  in  debt 
than  if  he  had  nothing?  Indicate  his  condition  by  the 
sign  -f  or  — . 

A  negative  quantity  is  sometimes  regarded  as  less  than  zero. 

6.  Two  vessels  left  the  same  port,  one  sailing  west  10 
miles  per  hour  and  the  other  sailing  east  8  miles  per  hour. 
How  far  were  they  apart  at  the  end  of  two  hours  ?  Indi- 
cate the  distance  sailed  by  each  vessel  and  the  direction. 


ADDITION. 


29.  1.    How  many  days  are  5  days,  4  days,  and  3  days  ? 

2.  How  many  cVs  are  o ri,  4 d,  3 fZ,  and  6d? 

3.  How  many  c's  are  6  c,  3  c,  5  c,  and  2c? 

4.  How  many  a&'s  are  3  ab,  2  a6,  4  a6,  and  5  a6  ? 

5.  When  no  sign  is  prefixed  to  a  quantity,  what  sign  is 
it  assumed  to  have  ? 

6.  When  positive  quantities  are  added,  what  is  the  sign 
of  the  sum  ? 

7.  If  Henry  owes  one  boy  3  cents,  another  5  cents,  and 
another  6  cents,  how  much  does  he  owe  ? 

8.  If  the  sign  —  is  placed  before  each  sum  that  he  owes, 
what  sign  should  be  placed  before  the  entire  sum  ? 

9.  What  sign  will  the  sum  of  negative  quantities  have? 

10.  If  a  vessel  sails  -f-  5  mi.,  +  8  mi.,  -|-  9  mi.,  and  is 
driven  back  —4  mi.,  —  2  mi.,  —  6  mi.,  how  far  is  she  from 
the  sailing  port  ? 

11.  A  boy's  financial  condition  is  represented  as  follows: 
Henry  owes  him  15  cents;  David  owes  him  10  cents  ;  James 
owes  him  8  cents.  He  owes  William  9  cents,  and  Fred  10 
cents.  What  is  his  financial  condition  ?  How  much  is 
15  cents,  10  cents,  8  cents,  —  9  cents,  —  10  cents  ? 

2Q 


ADDITION.  27 

30.  Principle.  Only  similar  quantities  can  be  united 
by  addition  into  one  term. 

Dissimilar  quantities  cannot  be  added,  but  in  algebra  an 
indicated  operation  is  often  regarded  as  an  operation  per- 
formed.    It  is  important  to  remember  that  fact. 

Thus,  a  and  h  cannot  be  added,  and  yet  a  +  6  is  called  their  sum, 
though  the  operation  is  only  indicated. 


31. 

To  add  similar  monomials. 

1. 

2. 

3. 

4. 

5. 

6. 

3aj 

9a& 

—  3  mn 

-  ^'y 

ax 

-      be 

Ix 

4.ab 

—  2  mn           - 

-5x^y 

^ax 

—    7  be 

X 

Sab 

—  9  mn           - 

-Sx'^y 

5  ax 

-    26c 

5ic 

6ab 

—     mn           - 

-*4.x-y 

4  ax 

-126c 

2x 

2ab 

—  5mn           - 

-Ixhi 

Sax 

-    6bc 

7.  Express  in  the  simplest  form   5a-|-3a  —  2a  —  7a  + 
12  a  +  3  a. 

Solution.  The  sum  of  the  positive  quantities  is  23  a,  and  the  sum 
of  the  negative  quantities  —  9a.  23a  —  9a  =  14a.  Hence  the  sum 
of  the  quantities,  or  the  simplest  form  of  the  expression,  is  14  a. 

Express  in  the  simplest  form  : 

8.  8a  — 2a +a  — 3o  —  a +  7a. 

9 .  4  x-y^  -h  3  x'-y^  —  x;-y-  -\-  5  xhf  —  10  xh/. 

10.  7  mx  -\-  4: 7nx  —  5  mx  —  2mx  —  ()  mx  +  mx. 

11.  oy —Sy-\-^y  — 10 y -\-6y  —  y. 

12.  Sm-^  Sm  —  om  —  2 m  -\-6m  —  4: m. 

13.  7  6c-h3  6c  — 4  6c  — 5  6c-i-8  6c  — 6c. 

14.  ^xy  -\-2xy  —  b  xy  -\- 10  xy  —  7xy  —  4t  xy  -\-  o  xy. 

15 .  Q>  x^z  —  4:  xh -\- S  x-z  +  8  x-z  —  5  xh  -f-  3  0^2;  —  10  xrz. 


28  ELEMENTS  OF   ALGEBRA. 

16.  15  mn  +  6  mn  —  10  mn  —  4  mn  —  3  mn  -f  4  m?i. 

17.  4  a2^>  _  3  a%  +  7  a-h  -  14  a-&  -  3  orb  +  20  a^^?. 
— 18.    25  aa;  —  17  a.f  —  13  aa?  -f- 19  ax  -|-  G  ax  —  20  ax. 

19.  3(a6)-  +  9(a&)--(a6)-  +  7(a&)--9(a6)l 

—  20.  8(a-6)+4(a-^)-6(a-6)-2(a-&)+3(a~6). 
21.  7  2/^2;  —  4  2/-2;  4- ?/-2;  —  6  ^2:  +  2  2/-2;. 

—  22.  5(a;+2/)-2(x-hy)-3(.^4-2/)+8(x+7/)-2(x4-2/). 
23.  4(a  +  6)2  +  10(a  +  ^)'-7(a  +  ^)2-2(a  +  6)2 

+  5(a  +  6)'. 
\  24.    3cd  —  2cd'\-bcd-\-lcd-3cd-^cd. 

25.    9(x//)'^  -  3{xyy  +  4(x?/)3  -  5(.x?/)-^  +  2{xyy  -  6{xy)\ 

32.   To  add  when  some  terms  are  dissimilar. 

1 .    Add  x  +  Sy  —  z,  x  —  2y,  x -\- 4: y  -}- 3  z. 

PROCESS.  Explanation.     Forconvenience  in  adding,  similar 

x  -\-  3y  —     z  terms  are  written  in  the  same  column,  and  the  sim- 

^.  _  2  »  plest  form  of  the  sum  is  obtained  by  beginning  at 

J          ,.  either  the  right  or  left  hand  column  and  adding 

^'  "»      y  ~r  *^  q^qy^  column  separately.     The  dissimilar  terms  in 

3x  4-  5y  4-  2z  ^^®  result  are  connected  with  their  proper  signs. 

^  Rule.  Write  similar  terms  in  the  same  column.  Add 
each  column  separately  by  finding  the  difference  of  the  sums  of 
the  positive  and  the  negative  terms.  Connect  the  results  with 
their  proper  signs. 

Find  the  sum  of  each  of  the  following : 

2.  3.  4. 

2a-46  10X  +  32/+    z  Sxy  +  2y^-    z' 

6 a  —  2  6  —  5  X  —    y  —  2  xy  -^  6  y^  —  5  z' 

2  a +  3  6  2x  —  2y-{-z  7  xy  —  4  y- -\- i  z' 

—  5  a  —  4  6  x-\-7y~lz  2y^  +  5z' 


ADDITIOX.  29 

5 .  Find  the  sum  of  2 c  -{-  5 d,  7c  —  d,  c?  —  4c,  2d  —  c. 

6.  Find  the  sum  of  6  7n  —  4:n,  2m-\-3n,  5?i  — 7m,  2n  — 3m. 
Express  in  the  simplest  form  : 

7.  2a4-2&-h3c  +  46-4a4-6a-2c. 

8.  xh  -\-  o  xz-  —  7  xy  -^  6  xz-  —  2  xh  +  4  a?i/  +  4  xh  —  xz^. 

9.  3w-^4:X  —  7y-^2r  —  2w—x-\-3y-\-4v  —  3x^4.W'-6v. 

10.  a'b'-\-c'-\-cd-2  c--3  cc?4-o  a^b'+cd-3  c--2  a'b'-c'. 

11.  Add  ab  +  a-c-5,3ab-3  a^c  -{-7,2  a^c- 2  ab  -  3. 

12.  Add  5a-\-3b  —  2c-{-d,2b-\-c-3d,7a-ob-\-c. 

13.  Add   6m+8n-f ir— ?/,   2m  — 2 n-f-3 0/^+41/,    — 7?i— 5a; 

+  2^. 

14.  Add  3x-{'7 y  —  4:Z  -{-6 w,  7 z  —  4:X  —  2y  —  oiv,  x-{-y 

-\-  Z  -{-  IV. 

15 .  Add  4:Xry  —  3  xy^  —  2  oi^y^,  4  xy^  —  3  x-y  —  2  x-y^,  4  a^2/^ 

—  2  x^y  —  3  xy\ 

16.  Add  3  aj"*  +  2  y'\  —  4a;"*  +  5 2/",  5  a;"'  —  4  ?/",  7  o.-"^  —  2 2/". 

17 .  Add     8  a-b'x'  —  3ab-\-  ed,    2  a-b'iv'  +  a6  —  4  ed,     2ab 

-  a'b"x\ 

18.  Add  7  m^  —  6  mn  -\-  5  ir,  4  ?;??i  —  3  m^  —  11^ ,  5  m^  —  4  n^ 

19.  Add     Uaa^-Say^-{-6az%     20ay^ -24:  ax^ -12  az^, 
32  aa.-^  -  40  ay'  +  15  a^^ 

20.  Add    10  a^b  -  12  a'bc- 15  b'c'-{- 10,     -  4:  a'b -{- S  a^bc 
_  10  bV  -  4,  2  a26  +  12  a'bc  -f-  5  6V  +  2. 

21.  Add     4a.'^-6aa^  +  5a2a;-5a^     3  a;^ -f- 4  aa.-^ -|- 2  a^a; 
-h  6  a^,  —  17  ar^  +  19  ax-  —  15  a^a;  +  8  a^. 

22.  Add  (j(c-{-d)  -3{c-{-d)  -\.a(c-{-d)  -26(c+d)-(c+d). 

23.  Add  7a  —  36-fc  +  m,  36  —  7a  — c  + m. 


30  ELEMENTS  OF  ALGEBRA.  | 

i 

24.  Add  Sax-^2{x  +  a)-\-3b,  9ax-{- 6{x -\- a) -9b,  \ 
llx-f  6b  —  7ax  —  H{X'j-a),  \ 

25.  Add6{x-{-y)+3z  —  S,2{x-\-y)-2z-^4:,Sz-3{x+y).  . 

I 

33.  A   letter   may  sometimes    represent    some    definite  ! 

number.  i 

Thus,  a  may  represent  5  ;  then  2  a  will  represent  10  ;  8  a,  15,  etc. 

A  letter  may  also  represent  any  number,  whatever  its  i 

value.  j 

Thus,  5  times  n  may  represent  5  times  any  number ;  8  times  n  or  | 
8  n  may  stand  for  8  times  any  number.  ; 

34.  Letters  used  to  represent  quantities  having  a  definite  : 
value,  or  letters  which  represent  any  number  or  quantity  are  : 
called  Known  Numbers  or  Quantities.  | 

The  Jirst  letters  of  the  alphabet,  as  a,  b,  c,  etc.,  are  used  j 

to  represent  known  numbers  or  quantities.  '      ] 

Thus,  a,  b,  c,  (?,  etc.,  are  usually  considered  known  quantities ;  ] 
that  is,  they  either  stand  for  known  numbers  or  for  any  number.  1 

PKOBI.EMS.  i 

35.  1.    A  man   bought   a  barrels   of   flour,  b  barrels  of  i 
sugar,  and  8  barrels  of  molasses.     How  many  barrels  in  all 
did  he  buy  ?  ] 

2.  Edith  bought  a  ribbon  for  m  cents,  a  pencil  for  d  ; 
cents,  and  a  book  for  6  cents.     How  many  cents  did  she  pay  ■ 

for  all?  "^  i 

1 

3.  A  farmer  sold  some  sheep  for  c  dollars,  a  cow  for  7i  1 
dollars,  and  a  horse  for  as  much  as  he  received  for  the  sheep  : 
and  cow.     How  much  did  he  receive  for  all  ?  ; 

4.  George  walked  a  miles,  he  then  rode  3  miles  on  his  i 
bicycle,  and  b  miles  on  the  cars.     How  far  did  he  travel  ?      ; 


ADDITIOX.  31 

5.  A  man  began  business  with  2c  dollars.  The  first 
year  he  gained  ^  as  much  as  he  had;  the  second  year  -^as 
much  as  he  had  at  the  end  of  the  first  year;  and  the  third 
year  $  400.     How  much  did  he  gain  in  the  three  years  ? 

6.  The  letter  h  represents  an  odd  number.  What  will 
represent  the  next  even  number  ?  What  the  next  odd 
number  ? 

7.  Laura  is  m  years  old ;  Lizzie  is  twice  as  old  as  Laura ; 
and  iNIabel's  age  is  equal  to  i  the  ages  of  the  other  two. 
W^hat  is  the  sum  of  their  ages  ? 

8.  A  merchant  took  in  c  dollars  one  week,  d  dollars  the 
next,  and  $  75  the  next.  How  many  dollars  did  he  receive 
in  the  three  weeks  ? 

9.  What  is  the  sum  of  x  -\-  x -\-  x  -\-  etc.  taken  seven 
times  ?     Oi  X -\-  X  -\-  X -{-  etc.  taken  a  times  ? 

10.  A  grocer  sold  h  pounds  of  sugar,  c  pounds  of  coifee, 
d  pounds  of  tea,  and  2  pounds  of  chocolate.  How  many 
pounds  of  groceries  did  he  sell  ? 

11.  A  man  paid  m  dollars  for  a  coat,  n  dollars  for  a 
waistcoat,  p  dollars  for  trousers,  g  dollars  for  a  pair  of 
boots,  and  r  dollars  for  a  hat.  How  much  did  his  outfit 
cost  him  ? 

12.  A  man  paid  a  dollars  for  a  farm ;  he  then  expended 
upon  improvements  d  dollars,  and  sold  it  for  h  dollars  more 
than  the  entire  cost.     How  much  did  he  receive  for  it  ? 

13.  My  fare  to  San  Francisco  was  a  dollars,  my  sleeping- 
car  charges  c  dollars,  my  meals  cost  me  h  dollars,  and  my 
other  expenses  $  25.  How  much  did  I  expend  before  I 
reached  San  Francisco  ? 


SUBTRACTION. 


36.  1.  What  is  the  difference  between  8  days  and  5 
days  ? 

2.  What  is  the  difference  between  11  cents  and  4  cents  ? 

3.  What  is  the  difference  between  12 d  and  5d? 

4.  What  is  the  difference  between  15c  and  8c? 

5.  What  is  the  diff'erence  between  12/ and  8/? 

6.  Subtract: 

Sx  from  13a? ;  4,y  from  l;")?/ ;  6  c  from  13c;  Sd  from  20 d; 
10a;^2/  from  14 a?^?/;  8afy  from  21  a^?/^;  4a7i/^  from  14 a??/-; 
10a^6^  from  18a^6^;  Sa^x-y-^  from  15 aV^/^;  Aabcd  from 
13a6cd;  6a6ca;  from  18a6ca7;  lOaaj^^;^  from  20axh/z^, 

7.  What  is  the  remainder  when   4:aVy^  is  subtracted 
from  10  a^x^y'^  ?    What  is  the  sum  of  10  aVy  and  —  4  a^x^y^  ? 

8.  What  is  the  remainder  when  3a6c^  is  subtracted  from 
12  ah(^  ?     What  is  the  sum  of  12  ahc^  and  -  3  ahc'  ? 

9.  Instead  of  subtracting  a  positive  quantity,  what  may 
be  done  to  secure  the  same  result  ? 

10.  What  is  the  result  when  8  is  subtracted  from  15? 
What,  when  8  —  5  is  subtracted  from  15  ? 

11.  Why  is  the  result  5  more  in  the  latter  case  than  in 
the  former  ? 

32 


SUBTRACTION.  33 

12.  What  is  the  result  when  IQa  is  subtracted  from 
18a  ?     What,  when  10a  —  8a  is  subtracted  from  18a  ? 

13.  Why  is  the  result  greater  by  8a  in  the  latter  case 
than  in  the  former  ? 

14.  Instead  of  subtracting  a  negative  quantity,  what  may 
be  done  to  secure  the  same  result  ? 

15.  One  vessel  was  40  miles  east  and  another  20  miles 
west  from  a  given  meridian.  Indicate  their  relations  by 
proper  signs.     How  far  apart  were  they  ? 

16.  Mr.  A's  property  is  worth  $  25a  and  Mr.  B  is  f  8a 
in  debt.  Indicate  their  financial  conditions  by  proper  signs. 
What  was  the  difference  in  their  financial  condition  ? 

17.  A  thermometer  indicated  a  temperature  of  35°  above 
zero  on  Jan.  5,  and  of  15°  below  zero  on  Jan  6.  Indicate 
the  temperature  by  proper  signs.  What  was  the  difference 
in  temperature  ? 

37.  Principles.  1.  The  difference  between  similar  quan- 
tities, only,  can  be  expressed  in  one  term. 

2.  Subtracting  a  positive  quantity  is  the  same  as  adding  a 
numerically  equal  negative  quantity. 

3.  Subtracting  a  negative  quantity  is  the  same  as  adding  a 
numerically  equal  positive  quantity. 

38.  To  subtract  when  the  terms  are  positive. 

1.  From  10  a  subtract  4  a. 

PROCESS.  Explanation.     When  four  times  any  number  is  taken 

-j  rw  from  ten  times  that  number,  the  remainder  is  six  times 

the  number  ;  therefore,  when  4  a  is  subtracted  from  10  a, 

"*  ^  the  remainder  is  6  a.     Or,  since  subtracting  a  positive 

^^— —  number  or  quantity  is  the  same  as  adding  an  equal  nega- 

^^         tive  quantity  (Prin.  2),  4  a  may  be  subtracted  from  10a 

by  changing  the  sign  of  4  a  and  adding  the  quantities.     Therefore,  to 

subtract  4  a  from  10  a,  we  find  the  sum  of  10  a  and  —  4  a,  which  is  6  a. 

milne's  el.  of  alg. — 3. 


34 


ELEMENTS  OF   ALGEBRA. 


2.    From  13  m  take  15  m. 


PROCESS. 

13m 
15  m 

-    2  m 


ExpLANATiox.  After  subtracting  from  13  m  as  much 
as  we  can  of  15  m,  there  will  be  2  m  yet  to  be  subtracted, 
or  the  result  will  be  —2  m.  Or,  since  subtracting  a  posi- 
tive number  or  quantity  is  the  same  as  adding  an  equal 
negative  quantity  (Prin.  2),  15  m  may  be  subtracted  from 
13  m  by  finding  the  suiA  of  13  m  and  —  15  m,  which  is 
—  2  m.    Therefore,  when  15  m  is  taken  from  13  m,  the 


result  is  —  2  m. 
3. 
From  19  a? 
Take     Ax 


4. 

7  ab 
Sab 


5. 

18  m2 
13  m^ 


6. 

16  xy 
20  xy 


7.  8. 

6  x^y-z  9  mn 

8  x^yh        14  mn 


Subtract  the  following : 

9.  8a;-|-32/ from  12a;-f72/. 

10.  5a6-f  3  c  from  10  a6  + 2  c. 

11.  4^>-f  9cZfrom  86  +  5d 

12.  5x-\-y  from  Sx-{- 3 y. 

13 .  4:X-{-Sz  from  5x-\-9z. 

14.  3a  +  2^  +  5c  from  7a +  5Z>  + 6c. 

15.  7a +  2&  from  9a -1-6. 

16.  5x'-\-Sy'--\-6xyhom7x^-^3y^  +  2xy. 

17.  4  a;?/ -f- 3  2;  from  10  a;?/ 4- 5  2. 

18.  5(a2  +  62)-h7c2-f-2d  from  6{a^ -\-b^)-\' 4.c^ +  4d. 

19.  3a(j9  +  g)  +  2  from  6a(p  +  g)+7. 

20.  5a;-h72/  +  42;-h5  from  5a7-h42/  +  82;-h6. 

21.  3x^-{-2y  -^z-\-Av  ivom  Tx^-^Sy  +  dz^v. 

22.  2a:-f-5(.v  +  2;)-h8  from  7x -\-2(y +  z)-\-10. 

23.  3a2  +  562  +  4c2  +  d2  ^^.^^  a^  j^Qb' +  c^ -\- 5d\ 


SUBTRACTION.  35 

39.  To  subtract  when  some  terms  are  negative. 

1.    From  7 x—2y  subtract  Ax—  3y. 

Explanation.     Since  the  subtrahend  is  composed 

PROCP.SS.      ^£  ^^^  terms,  each  term  must  be  subtracted  separately. 

7  X  —  2y  Subtracting  4 x  from  7  x  —  2  ?/,  leaves  ?jX  —  2y,or  the 

Ax  —  3v     ^^^^^^  ^^y  ^^  obtained  by  adding  —  4x  to  1  x  —  2y. 

But  since  the  subtrahend  was  3  y  less  than  4  x,  to  obtain 

the  true  remainder,  3 y  must  be  added  to  Sx  —  2y, 

3x  -\-    y     which  gives  ^x  -\-  y. 

Therefore  the  subtraction  may  be  performed  by 
changing  the  sign  of  each  term  of  the  subtrahend  and  adding  the 
quantities. 

Rule.  Write  similar  terms  in  the  same  column.  Change 
the  sign  of  each  term  of  the  subtrahend  from  -^  to  —,  or  from 
—  to  -\-,  or  conceive  it  to  he  changed,  and  proceed  as  in 
addition. 

2.  3.  4.  5.  6. 

YmmAa^x         9x^yz     6y-Az     7x^-\-5y'      3a-3b-{-c 
Take  3a^x     -'7x'yz     6y-\-2z     Ax^'-Sf-      3a  +  3h-c 


7. 

8. 

9. 

From 

Aah-3c 

+  d 

5a^- 

^3xy-^x 

10  ax  - 

-13y  + 

z^ 

Take 

-2a6-h5c 

-d 

6x^  -\-5xy  —  X 

5ax  - 

-  "^y- 

z" 

10.  From  12  ar^— 20  's^f  +  ^xy  subtract  9  :f?y^  —  2xy-^A^. 

11.  From  4 a6  +  3 6^  -  6 cd  subtract  (jab-2h--\-2 cd. 

12.  From  9  x^y  —  6  xy-  —  2xy-\-5  subtract  3  oi^y  -{-xy  -\-^. 

1 3 .  From  7  ic"*  -|-  2  x'^y''  —  5  y""  subtract  4  x""  —  2  x'^y''  —  9  y"". 

14.  From  8(m-f-?i2)-12(m2  +  7i)    subtract   12{m  ^  n-) 
-^(m'^n). 

15.  From  A:x^^7f-3z'-\-i^  subtract   -2a^ -\-y^-3r\ 

16.  From  b{p  -f-  q)  -  6(r  +  s)  -}-  15    subtract    8(p  +  q) 

+  2(r  +  60  +  25. 


36                         ELEMENTS   OF   ALGEBRA.  : 

17.  From  IS  m'nx^ -\- 12  a'bc"  ^  abd   subtract    ~  2  Tii^n:^^  I 
-I-  5  abd.  I 

18.  From  a^ -2ab -\- c^ -  3b^   subtract   2a'-2ab  +  3b'.  \ 

19.  From  2x'' -{-2y^  ^  4.xy   subtract   2 a^  +  2  ?/-  +  4  xy.  J 

20.  From  ^x^  -\-  Sx^y-  —  Sy^  subtract  the  sum  of  x'^  —  xry-  ' 
-f  3?/^   and   2x^ +  2x-y' -9y\  ] 

21.  From  5 x^y^ -\- 10 x^y  —  G yz^   subtract    lOa?^?/ —  4.ry'  '. 
H-  5yz^  ] 

22.  From  30;"^  — 4i»''i/'"-f  4i/"*   subtract   4  a;"^  +  2  x"?/"*  —  ?/^  ^ 

23.  From  Abx^  +  3  a?/-  +  4  —  c?/   subtract   cy  —  5  —  bx".  j 

24.  From  10?7i^  — 4m??  —  3n^  — 18   subtract  the  sum  of  \ 
m^  —  3 mn  +  4   and   5m-  —  2  mn  +  6 ti^.  i 

25 .  From  a^x  —  x^  -\-  x^y  —  8  subtract  3  a^x —5—xr-\-2  a%.  ; 

26.  From  150^2^-  — 15   subtract   AxY -{- z-  —  4y"  —  W.  i 

27.  From  2a;4-2/  +  :s  +  ?«    subtract   a?  +  2?/ -j- 2;$; -|-2?/. 

28.  F>om    16 a^b^— 12 be -\-14:b'-    subtract    the    sum    of  \ 
a'b'-3bc  +  A  and    10 a'b^ -J^S be  +  5 b\  ] 

29.  F^rom  ax  -\-  by   subtract   ex  —  dy, 

PROCESS.                    Explanation..    Since  a  and  c  may  be  re-  \ 

garded  as  the  coefficients  of  x,  and  6  and  -  d  \ 

ax  -f-  oy                            ^l^g  coefficients  of  y,  the  difference  between  i 

ex  —  dy                            the  quantities  may  be  found  by  writing  tiie  ; 

', r — r— -    difference  between  the  coefficients  of  x  and  //  ; 

(a      e)X  -f-(0  -\-  a)y    respectively  for  the  coefficients  of  tlie  remain-  ; 

der.    Since  c  cannot  be  subtracted  from  a,  the  \ 

subtraction  is  indicated  by  (a  —  c),  and  since  —  d  cannot  be  sub-  | 

tracted  from  6,  the  subtraction  is  indicated  by  (6  +  d),  consequently  ; 

the  remainder  may  be  written  (a  —  c)x  +  (6  +  d)y. 

30.  From  ax-^2y   subtract  2x  —  by, 

31.  'FTom2cx  —  ay  +  3z  subtract   cy-\-5x  —  az. 


SUBTRACTION.  37 

32.  From  ex —  12ab}j  +  ^aV   subtract   6x— 10 ahy-\- 3 b^. 

33.  From  2a(x  —  y)-\-4: abx   subtract   3c(x  —  y)-^2 ax. 

34.  From  ax  -^by  -{-  cz   subtract   bx  +  cy  +  dz. 

35 .  From  3x-\-7  y  -\-  Sz   subtract   box  ~  ay  -\-  az. 

36.  From  oay  -\-2cz  -{-  6x   subtract   cy  —  az  —  dx. 

37.  From   (a  —  b)x -\- {c -^  d)y   subtract   ax-\-dy, 

38.  From  7  a;^  -f-  5  ay^  -i-6z^  subtract   2  ax^  -f-  3  y-  —  abz^. 

39 .  From   ( m -f- n ) x^  -\-  ( m  —  n)y--\- z^   subtract    mno^ — 4 1/^ 

-f  az\ 

40.  From  «(.r  +  y)  -|-  ^(^'  —  Z/)  +  <^^  subtract  b(x  -^  y) 
—  a{x  —  y)  —  cx. 

41.  Express  the  difference  between  m  and  ?i. 

42.  Write  the  number  one  less  than  x\  the  number  one 
greater. 

.     43.    A  man  sold  a  horse  for  $125  and  gained  a  dollars. 
What  did  the  horse  cost? 

44.  A  girl  earns  b  cents  a  day  and  spends  25  cents  a 
week.     How  much  has  she  left  at  the  end  of  the  week  ? 

45.  The  sum  of  two  numbers  is  30,  and  x  represents  one 
of  them.     What  represents  the  other? 

46.  The  difference  between  two  numbers  is  5.  How  may 
the  numbers  be  represented  ? 

47.  A  merchant  bought  a  hat  for  b  dollars  and  a  coat 
for  c  dollars,  and  sold  the  two  for  d  dollars.  What  repre- 
sents his  gain  ? 

48.  A  man  whose  income  is  a  dollars  spends  m  dollars 
for  rent,  n  dollars  for  living  expenses,  and  100  dollars  for 
other  expenses.     What  represents  the  amount  he  saves? 


38  ELEMENTS  OF  ALGEBRA. 

49.  A  lady  paid  a  dollars  for  a  dress,  c  dollars  for  a  hat, 
and  $25  for  a  cloak.  How  much  had  she  left  from  a 
$50  bill? 

50.  A  man  paid  40  dollars  for  a  cords  of  wood,  and  sold 
it  at  3  dollars  a  corvl.     How  much  did  he  gain? 

51.  A  farmer  sold  some  grain  for  b  dollars,  some  fruit 
for  d  dollars,  and  some  hay  for  e  dollars.  He  received  in 
part  payment  a  horse  worth  /  dollars.  How  much  remained 
still  to  be  paid? 

40.  The  parenthesis,  (  ),  the  vinculum, ,  the  bracket, 

[  ],  and  the  brace,  \\,  are  called  Signs  of  Aggregation. 
They  show  that  the  quantities  indicated  by  them  are  to  be 
subjected  to  the  same  process. 

Thus,  (a  +  6)  X  c,  a  +  6  X  c,  and  {a  -h  h}  xc  show  that  the  sum  of  a 
and  h  is  to  be  multiplied  by  c. 

41.  The  subtrahend  is  sometimes  expressed  with  a  sign 
of  aggregation  and  written  after  the  minuend  with  the 
sign  —  between  them. 

Thus,  when  b  -\-  c  ~  d  is  subtracted  from  a  +  h,  the  result  is  some- 
times indicated  as  follows :    a  -\-  b  —(b  +  c  —  d)» 

1.  What  change  must  be  made  iu  the  signs  of  the  terms 
of  the  subtrahend  when  it  is  subtracted  from  the  minuend  ? 

2.  When  a  quantity  in  parenthesis  is  preceded  by  the 
sign  — ,  what  change  must  be  made  in  the  signs  of  the 
terms  when  the  subtraction  is  performed,  or  when  the  paren- 
thesis or  other  similar  sign  is  removed  ? 

The  term  parenthesis  is  commonly  used  to  include  all 
signs  of  aggregation. 

42.  Principles.  1.  A  j^arenthesis,  preceded  by  the  minus 
sign,  may  be  removed  from  an  expression  by  changing  the 
signs  of  all  the  terms  in  ptarenthesis. 


SUBTRACTION.  39 

2.  A  parenthesis,  preceded  by  the  minus  sign,  may  he 
used  to  inclose  an  expression  by  changing  the  signs  of  all  the 
terms  to  be  inclosed  in  parenthesis. 

When  quantities  are  inclosed  in  a  parenthesis  preceded  by  the  plus 
sign,  the  parenthesis  may  be  removed  without  any  change  of  signs, 
and,  consequently,  any  number  of  terms  may  be  inclosed  in  a  paren- 
thesis with  t\\e  plus  sign  without  any  change  of  signs. 

The  student  should  remember  that  in  expressions  like  —  (x"^  — ?/  +  2) 
the  sign  of  x^  is  plus,  and  the  expression  is  the  same  as  if  written 
-(+  x*^  -  ?/  +  z). 

Simplify  the  following: 

1.  20-(3-f4-6  +  o). 
Solution.    20  -  3  -  4  +  6  -  5  =  14. 

2.  2o-(6  +  9-3-7+13). 

3.  o-(3  +  2-6)-(-2-4-M). 

4.  (7-5  +  3)-(8-9  +  2). 

5.  lo-(3  +  4-6)-(8-6H-7). 

6.  (4 +  10 -8) -(3 +  7 -4). 

7.  19-(12-5-f  8)-(ll-5-3). 

8.  30-(-6  +  8-l)-(18-10  +  4). 

9.  (14-5  +  2)-(18-20  +  o). 

10.  lo-(3-f  8-7)  +  16-(10  +  4-f-o-8). 

11.  (5  +  7-8)-6-(-5  +  3-6  +  2). 

12.  16-[8-(5  +  6-4)-f-12]. 

Solution.       1(3  -  [8  -  (5  +  6  -  4)  +  12] 
=:  16  _  [8  -  5  -  6  +  4  +  12] 
-10-8  +  0  +  0-4-12 
=  3. 


40  ELEMENTS   OF   ALGEBRA. 

13.  25 -[13- 4 +  (3 -10 -2)]. 

14.  10- 58 -(15 +  7  + 3)  4- 6J. 

15.  17-(3  +  8)-[12-(3  +  8)-5]. 

16.  _  J- 16 +  13 -(6-1  4-4)4-5 -10|. 

17.  (3  +  7-4)-[14-(13  +  7)  +  5]. 

Simplify  the  following : 

18.  a  —  {b  —  c-{-d  —  e). 

Solution,     a  -{b  -  c  +  d  -e)  =a-b-^c-d  +  e. 

19 .  2x  —  {x  —  ox-\-3x  —  Sx). 

20.  7m  — (3n  4-2m  — 6m  4  >?.). 

21.  4  or  4-  7  ax  —  (5  ax  +  3  ax  —  2  oj-  +  10 ax). 

22.  a'-\-b'-{-2ab-2a'-2b'). 

23.  {6xy-{-2z)-{4.z-{-3xy-2z-{-5). 
~  24.  a  — 6  — (tt  +  6  —  c  — 3). 

25 .  a  4-  6  -  (2  a  -  3  6)  -  (5  a  4-  7  6)  -  (  -  13  a  +  2  6) . 

26.  (a  +  b-{-c)-{-{-a-\-b-c)-(:a-b-\-c), 

27.  x-l-x-^2x-{x-\-2x)-2x^. 

28.  3x  —  lx  —  3z  —  {2y  —  z)'], 

29.  a-  — a  — (4a  —  1/  — 3a2  — 1). 

30.  m  +  n  —  (m  4-  ^0  —  \  ^'^  —  ^^  ~"  (^^  +  ^0  —  ^^  • 

31.  {x'-]-2xy  +  y')-{2xy-x'-f). 


32.    9a;  — [8x  — 6ic  — 3a;]. 


33.  (a;4-10)-^,a;-3a;4-25-10|. 

34.  8a-(6a-5)  — (5a  +  ll-4a). 
36.    a-[26-(3c+26)-a]. 


SUBTRACTION.  41 

TRANSPOSITION  IN  EQUATIONS. 

43.  1.  If  a  certain  number,  diminished  by  3,  equals  15, 
what  is  the  number  ?     If  a;  —  3  =  15,  what  is  the  value  of  o^  ? 

2.  If  a  certain  number,  increased  by  3,  equals  15,  what 
is  the  number  ?     If  a;  -f  3  =  15,  what  is  the  value  of  a;? 

3.  In  the  equation  x  —  3  =  15,  what  is  done  with  the  3 
in  obtaining  the  value  of  a;?  In  the  equation  a;  =  15 +  3, 
how  does  the  sign  of  3  compare  with  its  sign  in  the  original 
equation  ? 

4.  In  the  equation  a;  -f-  3  =  15,  what  is  done  with  the  3 
in  obtaining  the  value  of  a;  ?  In  a;  =  15  —  3,  how  does  the 
sign  of  3  compare  with  its  sign  in  the  original  equation  ? 

5.  In  changing  the  3's  from  one  side  or  member  of  the 
equation  to  the  other,  what  change  was  made  in  the  sign  ? 

6.  When  a  member  or  quantity  is  changed  from  one 
member  of  an  equation  to  the  other,  what  change  must  be 
made  in  its  sign  ? 

7.  If  any  number,  as  5,  is  added  to  one  member  of  the 
equation  2  -f-  3  =  5,  what  must  be  done  to  the  other  member 
to  preserve  the  equality  ? 

8.  If  any  number,  as  3,  is  subtracted  from  one  member 
of  the  equation  2  +  3  =  5,  what  must  be  done  to  the  other 
member  to  preserve  the  equality  ? 

9.  If  one  member  of  the  equation  2  -f  3  =  5  is  multiplied 
by  any  number,  as  4,  what  must  be  done  to  the  other  mem- 
ber to  preserve  the  equality  ? 

10.  If  one  member  of  the  equation  2  +  3  =  5  is  divided 
by  any  number,  as  5,  what  must  be  done  to  the  other 
member  to  preserve  the  equality  ? 


42  ELEMENTS  OF   ALGEBRA. 

11.  If  one  member  of  the  equation  5  +  3  =  8  is  raised  to 
any  power,  as  the  second  power,  what  must  be  done  to  the 
other  member  to  preserve  the  equality  ? 

12.  What,  then,  may  be  done  to  the  members  of  an  equa- 
tion without  destroying  the  equality  ? 

44.  The  parts  on  each  side  of  the  sign  of  equality  are 
called  the  Members  of  an  Equation. 

45.  The  part  of  an  equation  on  the  left  of  the  sign  of 
equality  is  called  the  First  Member. 

c 

46.  The  part  of  an  equation  on  the  right  of  the  sign  of 

equality  is  called  the  Second  Member. 

47.  The  process  of  changing  a  quantity  from  one  member 
of  an  equation  to  another  is  called  Transposition. 

48.  A  truth  that  does  not  need  demonstration  is  called  an 
Axiom. 

Axioms.  1.  Things  that  are  equal  to  the  same  thing  are 
equal  to  each  other. 

2.    If  equals  are  added  to  equals,  the  sums  are  equal. 

'   3.    If  equals  are  subtracted  from  equals^  the  remainders  are 
equal. 

4.  If  equals  are  multiplied  by  equals,  the  products  are 
equal. 

5.  If  equals  are  divided  by  equals,  the  quotients  are 
equal. 

6.  Equal  powers  of  equal  quantities  are  equal. 

49.  Principle.  A  quantity  may  be  transposed  from  one 
member  of  an  equation  to  another  by  changing  its  sign  from 
+  to  — ,  or  from  —  to  -\-. 


SUBTRACTION.  43 

EQUATIONS  AND  PROBLEMS. 
50.     1.   Given  2  x  —  3  =  x  -\-  6,  to  find  the  value  of  x. 

PROCESS.  Explanation.     Since  the  known  and  the  un- 

o          r>        .   \   a  known  quantities  are  found  in  both  members  of 

^x  —  o  —  x  -f-  o  ^j^g  equation,  to  find  the  vahie  of  x,  the  known 

+  3  =     -{-3  quantities  must  be  collected  in  one  member  and 

7k                       r~^  the  unknown  in  the  other. 

^                      '  Since  —  3  is  found  in  the  first  member,  it  may 

X          =  x  be  caused  to  disappear  by  adding  3  to  both  mem- 

]          _  Q  bers  ( Ax.  2) ,  which  gives  the  equation,  2x  =  x-\-9. 

~~  Since  x  is  found  in  the  second  member,  it  may 

be  caused  to  disappear  by  subtracting  x  from 

^^'  both  members  (Ax.  3),   which  gives  as  a  result- 

2x  —  3  =  x  -\-6  ing  equation,  x  =  9. 

o^  _  J. fi-4-S  ^^'  since  a  quantity  may  be  changed  from  one 

~  member  of  an  equation  to  the  other  by  changing 

^=  "  its  sign  (Prin.),  —  3  may  be  transposed  to  the 

second   member  by  changing  it  to  +  3,   and  x 

VERIFICATION.  ^^^  ^^   transposed    to   the    first    member    by 

changing  it  to   —  x.     Then,  the  resulting  equa- 

18  —  3  =  9  +  6  tion  will  be  2x-ic  =  6  +  3. 

15  =  15  By  uniting  the  terms,  x  =  9. 

The  result  may  be  verified  by  substituting  the  value  of  x  for  x  in 
the  original  equation. 

If  both  members  are  then  identical^  the  value  of  the  unknown 
quantity  is  correct.  Thus,  if  9  is  substituted  for  x  in  the  original 
equation,  the  equation  becomes  18  —  3  =  9  +  6,  or  15  =  15. 

Therefore,  9  is  the  correct  value  of  x. 

Rule.  Transpose  the  terms  so  that  the  unknown  quantities 
stand  in  the  first  member  of  the  equation^  and  the  known 
quantities  in  the  second. 

Unite  similar  terms,  and  divide  each  member  of  the  equation 
by  the  coefficient  of  the  unknoicn  quantity. 

Verification.      Substitute    the    value    of   the    unknown 
quantity  for  the  quantity  in  the  original  equation.     If  both . 
members  are  then  identical  in  value,  the  value  of  the  unknoivn 
quantity  found  is  correct. 


44 


ELEMENTS   OF   ALGEBRA. 


Transpose  and  find  the  value  of  x. 


2.  0^  +  4  =  10. 

3.  .X-  — 3  =  4.  y 

4.  ^a;+l  =  o. 

5.  6x--G=  12. 

6.  4a.' +3  =  15. 

7.  8a;-2  =  14. 

8.  3a;+5  =  26. 

9.  9a;-5  =  3L 

10.  5i«  +  2  =  10  4-T. 

11.  7aj-l  =  30  +  4. 

12.  20? -10  =  3 +  5. 

13.  4x  — 2a;  =  3H-7. 

14.  6aj-3  =  2a;  +  9. 

15.  5a;— 15  +  2  =  2— 3a?4-l. 

16.  6a;- 2^  +  4  =  16-8. 

17.  7a;  — 3  =  2a;  — 4  4-11. 

18.  2a;  +  4  =  9a;-10. 

19.  6a;  +  25  =  18  — a;. 

20.  3a;-4  =  12-4. 

21 .  5  a;  —  5  =  67  —  3  a;. 

22.  10  a; -20  =  24 -12  a;. 

23.  3a;-14  =  10-.x. 

24.  2  a; -16  =  20 -4  a;. 

25.  15  a; -39  =  29 -2  a;. 

26.  3a;-18  =  31-4a;. 

27.  4a; -14  =  49 -3a;. 

28.  5a; -20  =  25 -4a;. 

29.  2a; -36  =  60 -6a;. 


30.  3  a;  — 20 -!-.«  =  44 -4  a;. 

31.  3a;  +  3  =  5  +  8.^;- 7. 

32.  9a;4-15  =  6  +  7a;  +  3. 

33.  5  a;  -f-  2  a;  =  9  a;  +  5  —  15. 

34.  8a; -10  =  10  + 2a;  4-4. 
35.-"ir^6a;-6  =  6  +  3-3a;. 

36.  x-:^  =  18  -  4  .f  -  3. 

37.  3a;  +  6a;  =1-8  -  a;  +  2. 

38.  3^— 6  =  a; +  14  — 4. 

39.  9a;+13=26  +  2.^;+l. 

40.  4a;  +  4- 3a;  =16-2.7;. 

41.  27  a; -14  =  190 -41a;. 

42.  ^x-  12=  4a; +  18. 

43.  5a;-  15  =  3 a; +9. 

44.  18a; +  9  =  15a; +  30. 

45.  7.T-3+2a;=5a;-20  +  l. 

46.  a;  +  14-2a;  =  6a;-21. 

47.  9a;  +  3-24  =  5.i;-25. 

48.  10;r -4 +  3=  6a;+ 19. 

49.  3a;-15-10  =  20-2.^•. 

50.  5  a;  — 5  — 20 +  6  a;  =  41. 

51.  2  a;  — 25  =  35  — a;  — 3  a;. 

52.  3a;-19  =  20-10.T+13. 

53.  5a;— 16  =  25— a;+40— 3a;. 

54.  6a;-5-30=10-4a;-5a;. 

55.  7a;-30  =  10  +  16-7a;. 

56.  5a;  — 50  =  25  — 5a; +25. 

57.  10a; -22  =  17 -2a; -a;. 


SUBTRACTION.  45 

PROBLEMS. 

51.  1.  Twice  a  certain  number  increased  by  15  is  equal 
to  the  number  increased  by  19.     What  is  the  number  ? 

2.  What  number  is  that  whose  double  exceeds  the  num- 
ber by  12  ? 

3.  Ten  times  a  certain  number  diminished  by  13  is 
equal  to  the  number  plus  o.     What  is  the  number  ? 

4.  What  number  diminished  by  8  equals  6? 

5.  Six  times  a  certain  number  plus  7  equals  five  times 
the  number  plus  12.     What  is  the  number  ? 

6.  Three  boys  together  had  90  cents.  The  first  had  10 
cents  more  than  the  second,  and  the  second  had  1  cent 
more  than  the  third.     How  much  had  each  ? 

7.  A  father  gave  a  certain  sum  to  his  youngest  son,  and 
4  cents  more  to  the  next  older,  and  10  cents  to  the  oldest. 
If  he  gave  to  all  20  cents,  how  much  did  he  give  to  each  ? 

8.  The  greater  of  two  numbers  exceeds  the  less  by  14, 
and  the  sum  of  the  numbers  is  34.    What  are  the  numbers  ? 

9.  A  and  B  started  in  business,  A  furnishing  $4000 
more  than  B.  Three  times  B's  capital  was  then  equal  to 
A's.    How  much  did  each  furnish  ? 

10.  A  and  B  had  the  same  sum  of  money.  A  gave  B  $4, 
and  then  B  had  double  the  amount  A  had  left^^— How  much 
had  each  at  first  ? 

11.  A  tourist  rode  32  miles  upon  a  bicycle.  A  certain 
number  of  miles  was  down  hill,  twice  as  far  plus  8  miles 
was  level,  and  the  distance  up  hill  was  ^  as  far  as  the  dis- 
tance on  a  level.  How  many  miles  did  he  travel  upon 
each  kind  of  road. 


46  ELEMENTS  OF  ALGEBRA.  j 

12.  A  man  has  two  horses,  of  unequal  value,  together  | 
worth  $200.  If  he  shouki  put  a  saddle  worth  $30  on  the  ; 
poorer  horse,  the  horse  and  saddle  would  together  be  equal  ■ 
in  value  to  the  better  horse  ?     What  is  the  value  of  each  ?  | 

13.  Six  hundred  gallons  of  water  are  discharged  into  a  ■ 
cistern  by  3  pipes.  The  second  discharges  100  gallons  j 
more  than  the  first,  and  the  third  discharges  three  times  as  j 
much  as  the  first.  How  many  gallons  are  discharged  by  each  , 
pipe  ?  I 

14.  A  drover  being  asked  the  number  of  his  cattle  said     ^ 
that  if  he  had  three  times  as  many  as  he  then  had  and  25 
more,  he  would  have  1000.     How  many  cattle  had  he  ?  f^r 

15.  A  man  bought  a  watch  and  chain  for  $60.  The  ! 
watch  cost  12  times  as  much  as  the  chain  lacking  $  5.  ■ 
What  was  the  cost  of  each?  .-   j 

16.  A  tenement  house  contained  90  persons,  men,  women,  \ 
and  children.     If  there  were  4  more  men  than  women,  and 
10  more  children  than  men  and  women  together,  how  many  ; 
were  there  of  each  ?  : 

17.  A  steamer  and  its  cargo  are  together  worth  $  120,000. 
If  the  steamer  lacks  only  $8400  of  being  worth  twice  as 
much  as  the  cargo,  what  is  the  value  of  each  ? 

18.  A  clerk's  expenses  are  $400  per  year,  and  his 
brother's  are  $600  per  year.  If  the  brother  has  three 
times  as  large  an  annual  salary  and  he  has  left  at  the  end  of 
the  year  a  sum  equal  to  twice  his  brother's  salary,  what  is 
the  salary  of  each  ? 

19.  If  a  house  and  lot  cost  four  times  as  much  as  the  lot, 
and  the  house  cost  $  2500  more  than  twice  as  much  as  the 
lot,  what  was  the  cost  of  each  ? 


MULTIPLICATION. 


^   52.    1.    What  is  the  sum  of  ^m-{-5m  -{-5m?     Or,  how 
much  is  3  times  5m?     8  times  bm"^ 

2.  What  is  the  sum  of  Sxy -\- %xy -\-%xy -{-^xy?  Or, 
how  much  is  4  times  ^xy?     10  times  ^xy? 

3.  How  much  is  6  times  1  he?  Which  quantity  is  the 
multiplier  ?  Which  is  the  multiplicand  ?  What  sign  has 
the  multiplier  ?  What  sign  has  the  multiplicand  ?  What 
sign  has  the  product  ? 

4.  When  a  positive  quantity  is  multiplied  by  a  positive 
quantity,  what  is  the  sign  of  the  product  ?    ^ 

5.  If  a  vessel  sails  south  8  miles  per  hour,  indicated  by 
—  8  mi.,  how  far  will  she  sail  in  5  hours  ?  What  will  be 
the  sign  of  the  product  ?    '•^-^ 

6.  How  much  is  4  times  —5xy?  6  times  —6ab? 
7  times  —Scd?  What  is  the  sign  of  the  multiplier  in 
each  case  ?  What  is  the  sign  of  the  multiplicand  ?  What 
is  the  sign  of  the  product  ?     —  4  a6  x  8  =  ?    —  Z  1,   "y   I 

7.  When  a  negative  quantity  is  multiplied  by  a  positive 
quantity,  what  is  the  sign  of  the  product  ?   — 

8.  How  does  the  product  of  6  times  7  compare  with 
the  product  of  7  times  6  ?  What  effect  upon  the  product 
has  it  to  change  the  order  of  the  factors,  when  the  numbers 
or  quantities  are  abstract  ? 

""'"■■'  47 


48  ELEMENTS  OF   ALGEBRA. 

9.  How,  then,  will  the  product  of  —  3  a  multiplied  by 
-j-  4  compare  with  the  product  of  -f-  4  multiplied  by  —  3a  ? 
What  is  the  product  ?  — 5a;2/x6— ?  6  x  —  o  xy  =  ? 
~7a6x4  =  ?     4.x-7ab=?^^  ^^^ 

10.  When  a  positive  quantity  is  multiplied  by  a  negative 
quantity,  what  is  the  sign  of  the  product  ? 

11.  How  much  is  6  times  —  3  a  ?^  2  times  —  3  a  ?  (6  —  2) 
times  —  3  a,  or  6  times  —  3  a,  —2  times  —3a? 

12.  Since.  2  times  —3  a,  or  —6  a,  must  be  subtracted 
from  —  18  a  to  obtain  the  correct  product,  what  will  be  the 
sign  of  —  6  a  after  it  is  subtracted  ?  -4~ 

13.  Since  —  2  times  —  3  a  gives  a  product  of  -{-6  a,  what 
may  be  inferred  regarding  the  sign  of  the  product  when  a 
negative  quantity  is  multiplied  by  a  negative  quantity  ? 

14.  What  is  an  exponent  ?  What  does  it  show  ?  What 
does  a^  mean  ?  When  w*  is  multiplied  by  a^,  how  many 
times  is  a  used  as  a  factor  to  obtain  the  product  ?  How 
many  times,  when  a^  is  multiplied  by  a^  ? 

16.    How,  then,  may  the  number  of  times  a  quantity  is 

used  as  a  factor  in  multiplication  be  determined  from  the 

exponents  of  the  quantities  in  the  expressions  multiplied  ? 

IHow  may  the  exponent  of  a  quantity  in  the  product  be 

determined  ? 


fe 


I  16.  3a2x6=?  10a"^x5  =  ?  20a2x3=  ?  25a^2/x2  =  ? 
I  How  is  the  coefficient  of  the  product  determined  from  the 
I  coefficients  of  the  factors,  or  from  the  multiplier  and  the 
I  multiplicand  ? 

63.    Multiplication  is  indicated  in  four  ways: 

1.   By  the  sign  x ,  read  multiplied  by  or  times. 
Thus,  a  X  h  shows  that  a  is  to  be  multiplied  by  b. 


MULTIPLIC  A  TION.  49 

2.  By  the  dot  (•),  read  multiplied  by  or  times. 
Thus,  a  '  b  shows  that  a  is  to  be  multiplied  by  b. 

3.  By  writing  letters,  or  a  number  and  a  letter  side  by 

side. 

Thus,  ab  shows  that  a  is  to  be  multiplied  by  b  ;  and  5  a  shows  that 
a  is  to  be  multiplied  by  6. 

4.  By  a  small  figure  or  letter,  called  an  Exponent,  written 

a  little  above  and  at  the  right  of  a  quantity,  showing  the 

number  of  times  the  quantity  is  to  be  used  as  a  factor. 

Thus,  a^  shows  that  a  is  to  be  used  as  a  factor  5  times,  or  that  it  is 
equal  to  «  x  a  x  a  x  a  x  a  or  aaaaa. 

54.  Principles.  1.  The  sign  of  any  term  of  the  product 
is  -f-  tvhen  its  factors  have  like  sig7is,  and  —  ivheji  they  have 
UNLIKE  signs. 

2.  The  coefficient  of  a  quantity  in  the  product  is  equal  to 
the  product  of  the  coefficients  of  its  factors. 

3.  The  exponent  of  a  quantity  in  the  proditct  is  equal  to  the 
sum  of  its  exponents  in  the  factors.       ^^^ 

55.  To  multiply  when  the  multiplier  is  a  monomial. 

1.  What  is  the  product  of  5  x-yz  multiplied  by  3  a6a;  ? 

PROCESS. 

Explanation.     The  coefficieut  of  the  product  is  ob- 
oxryz  tained  by  multiplying  5^y  3  (Prin.  2).     The  literal 

3abx  quantities  are   multiplied   by   adding   their  exponents 

(Prin.  3).     Hence,  the  product  is  I6abx^yz. 

15  abx^yz 

2.  What  is  the  product  of  3  6— ^.c  multiplied  by  JU5 V  ?       y 

PROCESS.  Explanation.     The  product  of  3  6  multiplied  by  * 

o,  5c2    is    15  6/j2^      But,    since    the    entire    multipli- 

~        _        cand  is  3  b^—  c,  the  product  of  c  multiplied  by  5  c^  - 
^^'        must    be  subtracted    from^l5  6c2.      The    product 
15  h(^  -__  5  (^  of  c  multiplied  by  bd^  is  5c^,  which  subtracted 

from  15  6c2  gives  the  entire  product  15  6c2  —  5  c^. 

milne's  el.  of  alg. — 4. 


50  ELEMENTS  OF   ALGEBRA. 

Rule.  Multiply  each  term  of  the  midtipUcanQl^  by  the 
multiplier,  as  folloivs  : 

To  the  product  of  the  numerical  coefficients,  annex  each 
literal  factor  ivith  an  exponent  equal  to  the  sum  of  the  expo- 
nents of  that  letter  in  both  factors. 

Write  the  sign  +  before  each  term  of  the  product  ivhen  its 
factors  have  like  signs,  and  —  ivhen  they  have  unlike  signs.  - 


3.        4.            6. 

6.          7.           8.          9.        10. 

Multiply 

10      10a    -6a 

22a;    18       -Ub    17c      24a; 

By 

4-4           3 

-5        6x^         3        3c    -9a; 

-/ 

11.           12. 

13.                 14.               15. 

Multiply 

5  oc^yz     —  15  xy^ 

16c^dz     -42mV     ~25a'b 

By 

2  x^yz     —    3  xy''' 
16.                  17. 

—  4:dz       —    3  m7i             5  a6^ 

18.                     19. 

Multiply 

(a-\-b)         4(x- 

■y)    3a;  4-2?/ -52;    2x-y-2y'z' 

By 

4         -5 

2a                 3a; 

Multiply : 

20.  3  a^a;  —  5  x-y  +  2  2/^  by  —  4  xy. 

21.  a^ -^2  ab  +  b- by  ab. 

22.  6  m  -t-  7  mn  +  5  n^  by  —  3  mn, 

23.  x^-2o^-\-5x^-}-x-3hy9x^ 

24.  9a^-lSab-^4:b^-6hjl2a^b', 

25 .  x^  —  Sxy  —  3xz-\-  yh^  by  2  ax. 

26.  4  a6  -f  3  a-b  -  5ab' -  2  a^hy  3  b\ 

27.  3  m^  —  10  mn —  8  n^  by  4  7n?i. 

28.  x'^-^x^-^-x^ -\-x-{-l\)y  —6x. 

29.  6a2-18a&-hl5c2-20a6c  +  14by3a252. 


MULTIPLICATION.  51 

56.   To  multiply  when  the  multiplier  is  a  polynomial. 

1.    Multiply  a-\-bhj  a-{-b. 

Solution.  a -\-  b 

a  -^  b 


a  times  a  -\-  b  =  a'^  -{-  ab 

b  times  a  -\-  b  —  ab  -\-  b^ 


(a  +  ?>)  times  (a  4-  b)  =  a*^  +  2  ab  +  b'^ 

Rule.      Multiply  each    term  of  the  midtiplicand  by  each 
term  of  the  multiplier,  and  add  the  partial  products. 


2. 

2ab-Zc 
4ab  -}-  c 

-3c-2 

3. 

X  -y 

Sa%'-  \2abe 
2  abc  - 

x^-S:x^y-\-Sx-^yi-xy^ 

-    x^y  +  SxY-Sxy^-{-y^ 

8  aVy^  -  10  abc  - 

-3c2 

x*  -ix^y-^G x'hf  -  4  xy^  +  y^ 

Multiply  : 

4.  x-\-y  hy  X -\-y.  17.  5m  —  4:7i  hj  Am -{- oy. 

5.  m  4-  ^i  by  m  —  n.  18.  x-\-2yhyx-{-5  y. 

6.  a--^2cdj-\-b-hy  a-\-b.  19.  1 -\- x -{- x^- by  1  —  x. 

7.  2  a  — 5  6  by  2  a -h  5  6.  20.  x -{- y -\-lhy  x  —  y —  1. 

8.  aj-f4byx  — 10.  21.  2a7 +  42/ by  3ic  —  2i/. 

9.  3y-^2zhy  2y-\-3z.  22.  a  +  6  -  2c  by  2a  -  6. 

10.  3  a +  76  by  3  a  — 76.  23.  4  a;  +  7  by  3  a:  — 2. 

11.  2ic  +  l  by  3a.-  — 6.  24.  3  am  +  6c  by  8ac  +  cl 

12.  2a  — 36  by  3a +  56.  25.  3 a;?/ —  6 ?/ by  4 iui/ +  8 y. 

13.  3??i  +  47?  by  2m  +  3?^  26.  a;  +  ?/ —  2;  by  a;  +  ?/. 

14.  5?/ —  32?  by  4?/ —  4^;.  27.  a  — 6  — cbya  — c. 

15.  26  — 5c  by  36+ 8c.  28.  2a -{- x  —  y  by  a  —  x. 

16.  3a; -20  by  8a; +  4.  29.  2x -\- 3y  —  6  by  x -\- oy. 


52  ELEMENTS   OF   ALGEBRA. 

30.  3xz-\-2y^hy  Sxz-4:y\ 

31.  7n  -f  ?i  +  1  by  m  —  n  +  1. 

32.  ox-{-4:hj5x—  9. 

33.  3 XT  —  4:y- by  3 or  ^7 yi 

34.  cf  -f-  3  d'h  +  3  ah-  +  W  by  a  -|-  Z>. 

35.  4ar  —  12ii7?/-f-9/ by  2i»  — 3?/. 

36.  m^  +  7/i'7i  -j-  mhr  -f  wi^i^^  4-  ^^^  by  y>i  —  ?i. 


/ 


PROBLEMS. 

57.  1.  If  from  three  times  a  number  4  is  subtracted  and 
the  remainder  is  multiplied  by  6,  the  result  is  12.  AV'hat 
is  the  number  ? 

2.  If  from  two  times  a  number  4  is  subtracted  and  the 
remainder  is  multiplied  by  3,  the  result  equals  two  times 
the  sum  of  that  number  and  2.     What  is  the  number? 

3.  A  father  is  four  times  as  old  as  his  son,  and  5  years 
ago  he  was  seven  times  as  old  as  his  son.  What  is  the  age 
of  each  ? 

4.  Samuel  and  John  together  have  40  cents.  If  John 
had  5  cents  less,  and  Samuel  5  cents  more,  Samuel  would 
have  three  times  as  much  money  as  John.  How  many  cents 
has  each  ? 

6.  A  commenced  business  with  three  times  as  much 
capital  as  B.  During  the  first  year  A  lost  \  of  his  money, 
and  B  gained  $500.  The  amount  of  A's  and  B's  money 
was  then  equal.     How  much  had  each  at  first  ? 

6.  A  is  50  years  of  age ;  B  is  10.  When  will  A  be  three 
times  as  old  as  B  ? 

7.  Six  men  hired  a  boat,  but  2  of  them  being  unable  to 
pay  their  share,  the  other  4  were  obliged  to  pay  1  dollar 
more  each.     For  how  much  did  they  hire  the  boat  ? 


MULTIPLICATION.  63 

8.  Three  times  the  difference  between  a  certain  number 
and  10  equals  two  times  the  sum  of  the  number  and  10. 
What  is  the  number  ? 

9.  Express  the  product  of  the  factors  2,  x,  y,  z,  x^,  y,  Az, 

10.  What  will  d  quarts  of  milk  cost  at /cents  per  quart? 

11.  How  far  will  a  man  travel  in  a  hours  if  he  goes 
b-\-6  miles  per  hour? 

12.  A  farmer  has  a  cows  and  three  times  as  many  sheep 
less  8.     How  many  animals  does  he  own  ? 

13.  A  man  sold  20  acres  of  land  at  a  dollars  per  acre. 
With  a  part  of  the  money  he  bought  3  horses  at  d  dollars 
each.     How  much  money  had  he  left  ? 

14.  If  a  men  can  do  some  work  in  12  days,  how  long  will 
it  take  one  man  to  do  the  same  work  ? 

15.  A  starts  in  business  with  b  dollars;  B  starts  with 
c  dollars.  In  one  year  A  gains  as  much  more,  while  B 
gains  i  as  much  more.  How  much  has  each  at  the  end  of 
the  year  ? 

16.  What  will  10  bushels  of  potatoes  cost  at  2  m  cents 
per  bushel  ? 

17.  A  man  earns  f  2  per  day  and  pays  fa  per  week  for 
his  board.  How  much  money  will  he  have  at  the  end  of 
b  -weeks  ? 

18.  An  engine  pumps  150  gallons  of  w^ater  into  a  tank 
each  day  ;  10  c  gallons  are  drawn  off.  How  much  water  will 
remain  in  the  tank  at  the  end  of  4  days  ? 

19.  The  daily  w^ages  of  a  mechanic  are  a  dollars.  How 
much  will  the  wages  of  10  mechanics  for  c  days  be  ? 


64  ELEMENTS  OF   ALGEBRA. 

SPECIAL  CASES  IN  MULTIPLICATION. 

58.    The  square  of  the  sum  of  two  quantities. 

a  -f       b 
a  -f       b 


m 

+ 

n 

-f 

m~ 

+ 

mn 
mn 

n^ 

71V 

4-  2  mil 

+ 

n- 

a^  -\-    ab 

ab  +  b^ 

a-  +  2  a6  +  &^ 

1.  How  is  the  first  term  of  the  second  power,  or  square,  j 
of  the  quantities  obtained  from  the  quantities  ?  How  is  ] 
the  second  term  obtained  ?     The  third  term  ? 

2.  What  signs  have  the  terms  ? 

59.  Principle.  The  square  of  the  sum  of  two  quantities 
is  equal  to  the  square  of  the  first  quanilty,  plus  twice  the  pro- 
duct of  the  first  and  second,  plus  the  square  of  the  secoyid. 

Write  out  the  products  or  powers  of  the  following : 

1.  {x-\-y)(x-{-y).  13.    Square  2 a; -f- 5. 

2.  (6+c)(6  +  c).  14.    Square  3m +  1. 

^    3.  (m  +  -)(m  +  2).  16.  Square  2a  +  5&. 

4.  {a  -{-x){a  -\-x).  16.  Square  a^  +  6-. 

5.  (if  4- 3)  (a; +  3).  17.  Square  aj^  +  3. 

6.  (?/-|-l)(2/  +  l).  18.  Square  2  m' -f  3  nl 

7.  {2x  -^y){2x-^y).  19.  Square  a6 -h  2  c. 

8.  {m-\-2n){m -\-2n).  20.  Square  2 icy  +  2;. 

9.  (3a  +  &)(3a4-^).  21.  Square  2/^  +  4 2;l 

10.    (2i«-f  32/)(2a;  +  32/).  22.    Square  0^  +  8.  j 


11.  (a +  4  6)  (a +  4  6).  23.    Square  5  a -f  "  &• 

12.  (2m+2M)(2m+2n).         24.    Square  4 a^ -j- 3 &-. 


MULTIPLICATION^. 


55 


60.   The  square  of  the  difference  of  the  two  quantities. 


X  —      y 

c  -      d 

X  -      y 

c  —      d 

or—    xy 

c-  —     cd 

—    072/  4-  r 

-    cd-\-d' 

x'^2xy-{-  7/ 

c}-2cd^d' 

1.  How  is  the  first  term  of  the  second  power  obtained 
from  the  terms  of  the  quantity  squared  ?  How  is  the  second 
term  obtained  ?     The  third  term  ? 

2.  What  signs  connect  the  terms  of  the  power  ? 


61.  Principle.  The  square  of  the  difference  oftioo  quan- 
tities is  equal  to  the  square  of  the  first  quantity,  minus  twice 
the  product  of  the  first  and  second,  plus  the  square  of  the 
second. 

Write  out  the  products  or  powers  of  the  following : 


1. 

(a  —  x){a  —  x). 

13. 

Square  2a- 3  6. 

2. 

{h-c){h-c). 

14. 

Square  m  —  2n. 

3. 

{m  —  7i)(m  —  >i). 

15. 

Square  2  6  -  4  d 

4. 

{x-2)(x-2). 

16. 

Square  a^  -  2  h\ 

5. 

(y-z){y-z). 

17. 

Square  he  —  xy. 

6. 

(a-;U)(a-3  6). 

18. 

Square  2x^  —  o  y-. 

7. 

(b-2c){b-2c). 

19. 

Square  2a  —  c. 

8. 

{2x-2y){2x-2y), 

20. 

Square  3m^—  1. 

9. 

{b-o){b-5). 

21. 

Square  3  mn  —  4. 

10. 

(y-l){y-l). 

22. 

Square  y^  —  6. 

11. 

{ah -2)  {ah -2). 

23. 

Square  Ax^  —  by\ 

12. 

(a; -4)  (a; -4). 

24. 

Square  ah  —  2  (?. 

56  ELEME^v^TS   OF   ALGEBRA. 

62.   The  product  of  the  sum  and  difference  of  two  quantities. 


X  -    y 
X  -f    y 


x^  —  xy 


a;y  -  y' 


x" 


-r 


c  -{-  d 
c  —   d 

c^  +  cd 
-  cd  -  d^ 


1.  How  are  the  terms  of  the  product  of   the  sum  and 
difference  of  two  quantities  obtained  from  the  quantities  ? 

2.  What  sign  connects  the  terms  ? 


63.     Principle.     The  product  of  the  smn  and  difference 
of  tivo  quantities  is  equal  to  the  difference  of  their  squares. 

Write  the  products  of  the  following : 


1. 

{a-{-b){a-b). 

13. 

(b  +  2c){b-2c). 

2. 

(m  4-  >i)  (^>i  —  n). 

14. 

(Sx-\-8y)(3x-8y), 

3. 

(a-^x){a  —  x). 

15. 

(a; +  10)  (a.- -10). 

4. 

(2a-\-b)(2a-b). 

16. 

{bc-\-ef){bc-ef). 

6. 

(2x-\-y)(2x-y), 

17. 

(Sx'-{-2f)(3^--2f), 

6. 

(a +  4)  (a -4). 

18. 

(5a  +  3a;)(5a-3i^0- 

7. 

(2m+37i)(2m-3n). 

19. 

(a^  +  b')(a'-b'). 

8. 

(y-^l)(y-l). 

20. 

(mn  +  4)(?/iyi  — 4). 

9. 

(x  +  5)(x-o). 

21. 

(x  +  6){x-6). 

10. 

(2  +  y)(2-2/). 

22. 

(4:y-\-7z)(^y-7z). 

11. 

(ab-\-3c)(ab-3c). 

23. 

(3x-^^y){Sx-4y). 

12. 

(2m4-2n)(2m~2n). 

24. 

{2ab+5c)(2ab-'5c). 

MULTIPLICATION.  57 

64.    The  product  of  two  binomials. 

X  +3  a;  +3  x  —3 

X  4-5       "  X  ^  5  X  —  5 


x^-i-3x  xF4-3x  x^-3x 

5a;  +  15  —  5a;  —  15  —  5  x  +  15 

a;2-f  8a;-fl5  ar'_2a;-15  x'-8x-{-15 

1.  How  is  the  first  term  of  each  product  obtained  from 
the  factors? 

2.  How  is  the  second  term  of  the  product  in  the  lirst 
example  obtained  from  the  factors?  The  second  term  in  the 
second  example?     The  second  term  in  the  third  example? 

3.  How  is  the  third  term  of  the  product  in  each  example 
obtained  from  the  factors  ? 

4.  How  are  the  signs  determined  which  connect  the 
terms  ? 

65.  Principle.  The  product  of  two  binomial  quantities 
having  a  common  term  is  equal  to  the  square  of  the  common 
term,  the  algebraic  sum  of  the  other  two  multiplied  by  the 
common  term,  and  the  algebraic  product  of  the  unlike  terms. 

Write  the  products  of  the  following : 


1. 

(a; +  3)  (a; +  4). 

8. 

(aj_4)(a;-f  8). 

2. 

{x-l){x^5). 

9. 

(a; +  9)  (a; +  3). 

3. 

{x-2){x-3). 

10. 

(a; -12)  (a; +  6) 

4. 

(x  +  %){x-l). 

11. 

{x-5){x-l). 

6. 

(a;  +  5)(a;-f  10). 

12. 

(a;  +  14)(;i;-4) 

6. 

(^-13)  (a; -4- 3). 

13. 

(,;_!)  (^4. 8). 

7. 

(a;-F20)(a;  +  5). 

14. 

(a; -5)  (a; -4). 

58  ELEMENTS  OF   ALGEBRA. 

16.  (x-\-ll)(x-2).  20.  (a; 4-9)  (a? 4- 12). 

16.  (x-25)(x-4.).  21.  (a; -10)  (a;  4- 12). 

17.  (a; +  5)  (a;  4- 15).  22.  (a;  —  2.^)  (a;  4- 47/). 

18.  (a;  — 6)(a.'  — 3).  23.  (a;  —  a)  (a;  —  7a). 

19.  (aj4-6)(a;-3).  24.  (x -{- 6 y)  {x -\- 10 y) . 

SIMULTANEOUS  EQUATIONS. 

66.  1.  If  the  sum  of  two  numbers  is  8,  what  are  the 
numbers  ?  How  many  answers  may  be  given  to  the  ques- 
tion ? 

2.  Let  X  and  y  stand  for  the  two  numbers;  then,  in  the 
equation  x  -\-  y  =  S,  how  many  values  has  x?  How  many 
has  y?  How  many  values  has  each  unknown  quantity  in 
such  an  equation  ? 

3.  In  the  equation  x -^  y  =  20,  what  is  the  value  of  y,  if 
x  =  S?     Ifa.'  =  6?     Ifa;  =  4?     Ifaj  =  12?     If  a;  =  10? 

4.  If  the  equations  x-\-y  =  6  and  x  —  y  =2  are  added 
together  (Ax.  2),  what  is  the  resulting  equation?  What 
is  the  value  of  x  in  these  equations  ?     Of  y? 

^  67.    Equations  in  which  the  same  unknown  quantity  has 
the  same  value  are  called  Simultaneous  Equations. 

68.  The  process  of  deducing  from  simultaneous  equations 
other  equations  containing  a  less  number  of  unknown  quan- 
tities than  is  found  in  the  given  equations,  is  called  Elimina- 
tion. 

69.  Elimination  by  addition  or  subtraction. 

1.  If  the  equations  x -{-3y  =  9  and  x  —  3y  =  3  are  added, 
what  is  the  resulting  equation  ?  What  quantity  is  elimi- 
nated by  the  addition  ? 


SIMULTANEOUS  EQUATIONS.  59 

2.  How  may  the  equations  2  x  —  4  ?/  =  4  and  ic  +  4  ?/  =  8 
be  combined  so  as  to  eliminate  y  ? 

3.  How  may  the  equations  3  a;  +  4  ?/  =  18  and  3  iK  +  ?/  =  9 
be  combined  so  as  to  eliminate  x  ? 

4.  When  may  a  quantity  be    eliminated   by    addition  ? 
When  by  subtraction  ? 

5.  \i  x-\-'6y  =  o  and  2 x  +  3 ?/  =  7,  how  may  the  values 
of  X  be  found  ? 

6.  If3a;  —  2/  =  5  and  2x-{-  y  =  6^  how  may  the  value  of 
X  be  found  ? 

70.  Principle.  Quantities  may  he  eliminated  by  addition 
or  subtraction  when  they  have  the  same  coefficients. 

71.  1.  Find  the  value  of  x  and  y  in  the  equations  it*  -|-  3  y 
=  11  and  2a.-- 4  7/ =  2. 

.,           ..  ..  Explanation.      Since   the   quantities 

X  -\-f:y  y  =  II  \i)  \l^^^f^i  not  the  same  coefficients,  we  must 

2  a;  —  4  7/  =  2  (2)  multiply  the  equations  by  such  numbers 

~          ~          ~  Q.  as  will  make  the  coefficients  alike.     If 

4  0/*  -f-  Jw^^  =  44  (o)  ^g  ^jgj^  ^Q  eliminate  ?/,  we  must  multiply 

6  a;  —  12  :y  =  6  (4)  (i)  by  4  and  (2)  by  3  (Ax.  4).    We  may 

TT~       ~  ^.  now  eliminate  y  from  equations  (3)  and 

10  a;  =  oU  (O)  ^4>j    |3y   addition.      From    the    resulting 

X  =  5  (6)  equation,    10  x  =  50,   the   value   of   x  is 

.,  obtained  by  dividing  each  member  by  10, 

5  +  o  //  =  11  (0  ^ijg  coefficient  of  x. 

3  y  =  i)  (8)  By  substituting  the  value  of  x  in  equa- 

^ /  _  2  (9)  tion  (1),  equation  (7)  is  obtained,  and  the 

^  value  of  y  is  found  to  be  2. 

EuLE.  If  necessary,  multiply  one  or  both  equations  by  such 
a  quantity  as  ivill  cause  one  unknown  quantity  to  have  the  same 
coefficient  in  each  equation. 

When  the  signs  of  the  equal  coefficients  are  alike^  subtract 
one  equation  from  another ;  when  the  signs  are  unlike^  add  the 
equations. 


60  ELEMENTS  OF   ALGEBRA.  ; 

i 
Find    the   values   of    the    unknown   quantities    in    the 
following  equations : 


2. 


3. 


5. 


}     X-    y=    2}  '     \ox-j-:Uj=]6) 

c    x-2y=    4|.  ^^      ^9x-^-ij/=      31^ 

\2x-    y  =  lli    .  '     i2a;-8y  =  -    2)" 

I    a:-f-3y  =  17|  ^^      |2?/+    z  =  26) 

\2x-2ij=    2)  '     \27j-\-2z  =  2S^ 

f    ^4-    l/=    4|  ^^      <2^  +  3.  =  23| 

(4a;-    ?/=    1)  '     \3y-2z=    2) 

(2a; +  2^  =  20)  ^^'      (3a'+      ;i  =  18  I 

(2.T-    //=    3 1  ^g^      (     a;  4- 2// =  30  I 

^      (    ..  +  2,v=:23|  ^^      r3.r-    ./=      30 1 

l3.b'+    2/=34i  *     1    ;«-3//  =  -30) 

(  4.1-  +  2//  =  14  I  (  2a;  +  3//  =  14  ^ 


7. 


^//=14| 
y=    43 


21. 


(9a;-    y=    4  3  (3a;  +  2/y=16  3 

10.     j^^^-^y  =  lM  22.     1^^'^-     '  =  ^^1 

(     a;+    yz=    7)  (l4?/-3;2=    3) 

^^      ^2x-    5y=.    6|  ^^       (5..  +  2,=  6| 

(ox-  \2y  =  16  3  ( 4a;  +  3 //  =  2  i 

12.     }     ^+    ^=^n  24.     }6-3.=  12) 

l79;-2y=   9)  (ox-    2/=13f 

l6a;-oy=    2J  '     l3.r-2y  =  16i 


SIMULTANEOUS  EQUATIONS.  61 

PROBLEMS. 

72.  1.  The  sum  of  two  numbers  is  10,  and  their  differ- 
ence is  2.     What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  14,  and  the  greater  plus 
two  times  the  less  is  20.     What  are  the  numbers  ? 

3.  A  and  B  together  have  $300.  Three  times  B's  money 
added  to  five  times  A's  gives  $  1100.  How  much  money 
has  each  ? 

4.  The  sum  of  the  ages  of  a  father  and  son  is  50  years. 
The  difference  between  the  father's  age  and  two  times  the 
son's  is  20  years.     What  is  the  age  of  each  ? 

5.  A  boy  has  25  marbles  in  two  pockets.  Twice  the 
number  in  one  pocket  equals  three  times  the  number  in  the 
other.     How  many  marbles  has  he  in  each  pocket  ? 

6.  A  farmer  paid  $3400  for  100  acres  of  land.  For 
part  of  it  he  paid  $  30  per  acre,  and  for  part  of  it  $  40  per 
acre.     How  many  acres  were  bought  at  each  price  ? 

7.  A  boy  spent  35  cents  for  oranges  and  pears,  buying 
in  all  13  oranges  and  pears.  He  paid  3  cents  apiece  for  the 
oranges  and  2  cents  apiece  for  the  pears.  How  many  of 
each  kind  did  he  buy  ? 

8.  Two  men  start  in  business  with  $  5000  capital. 
Twice  the  amount  B  furnishes  taken  from  twice  the  amount 
A  furnishes  will  leave  the  amount  B  furnishes.  What 
cajjital  does  each  furnish  ? 

9.  There  are  two  numbers  such  that  five  times  the 
first  minus  three  times  the  second  equals  4,  and  two  times 
the  first  plus  the  second  equals  17.     What  are  the  numbers  ? 

10.    A  farmer  sold  5  horses  and  7  cows  to  one  person 
for  f  745.     To  another  person  at  the  same  price  per  head 


62  ELEMENTS  OF   ALGEBRA. 

he  sold  3  horses  and  10  cows  for  $650.     What  was  the 
price  per  head  of  each  ? 

11.  A  man  and  wife  working  for  6  days  received  $  15. 
Again,  the  man  worked  for  4  days  and  the  wife  5  days,  and 
they  received  $  11.     What  were  the  daily  wages  of  each  ? 

12.  The  sum  of  two  numbers  is  26.  The  iirst  minus 
twice  the  second  is  8.     What  are  the  numbers  ? 

13.  A  purse  contained  $  30  in  one  and  two  dollar  bills. 
If  the  whole  number  of  bills  was  18,  how  many  bills  were 
there  of  each  kind  ? 

14.  A  merchant  sold  2  yards  of  velvet  and  4  yards  of 
broadcloth  for  $  16.  Again  he  sold  3  yards  of  velvet  and 
5  yards  of  broadcloth  for  $  21.  What  was  the  price  of  each 
per  yard  ? 

15.  If  A  gives  B  f  5  of  his  money,  B  will  have  twice  as 
much  money  as  A  has  left;  but  if  B  gives  A  f  5  of  his 
money,  A  will  have  three  times  as  much  as  B  has  left.  How 
much  money  has  each  ? 

16.  A  boy  who  desired  to  purchase  some  writing  pads 
and  some  pencils  found  that  2  pads  and  7  pencils  would 
cost  him  31  cents,  and  that  3  pads  and  4  pencils  would  cost 
27  cents.     What  was  the  price  of  each  ? 

17.  The  wages  of  10  men  and  8  boys  per  day  were  $28, 
and  the  wages  of  7  men  and  10  boys  at  the  same  rate  were 
$  24.     What  were  the  daily  wages  of  each  ? 

18.  A  man  received  at  one  time  $  17  for  sawing  8  cords 
of  wood  and  splitting  10  cords,  and  at  another  time  f  13.50 
for  sawing  5  cords  of  wood  and  splitting  12  cords  at  the 
same  rates  as  on  the  former  occasion.  What  did  he  receive 
per  cord  for  the  sawing  and  for  the  splitting? 


DIVISION. 


73.    1.  Since  +5  multiplied  by   +4  is  +20,  if  -f  20  is 
divided  by  +  o,  what  is  the  sign  of  the  quotient? 

2.  What,  then,  is  the  sign  of  the  quotient  when  a  posi- 
tive quantity  is  divided  by  a  positive  quantity  ? 

3.  Since  -\- o  multiplied  by  -■  4  is  —20,  if  —20  is 
divided  by  -f  5,  what  is  the  sign  of  the  quotient  ? 

4.  What,  then,  is  the  sign  of  the  quotient  when  a  nega- 
tive quantity  is  divided  by  a  positive  quantity  ? 

5.  Since  —5  multiplied  by  -f-4  is  —20,  if  —20  is 
divided  by  —  5,  what  is  the  sign  of  the  quotient  ? 

6.  What,  then,  is  the  sign  of  the  quotient  when  a  nega- 
tive quantity  is  divided  by  a  negative  quantity  ? 

7.  Since  —5  multiplied  by  —4  is  -f  20,  if  -f- 20  is 
divided  by  —  5,  what  is  the  sign  of  the  quotient  ? 

8.  What,  then,  is  the  sign  of  the  quotient  when  a  posi- 
tive quantity  is  divided  by  a  negative  quantity  ? 

9.  What  is  the  sign  of  the  quotient  when  the  dividend 
and  the  divisor  have  like  signs  ?  What,  w^hen  they  have 
unlike  signs  ? 

10.  How  many  times  is  6  x  contained  in  12  .r  ?  8  y  in  24y  ? 

11.  How,  then,  is  the  coefficient  of  the  quotient  found? 

63 


64  ELEMENTS   OF   ALGEBRA. 

12.  Since  x^  x  x^  =  .^•^  if  x^  is  divided  by  a--,  what  is  the 
quotient  ?     What,  when  x^'  is  divided  by  x^  ? 

13.  Since  a^  x  a^  —  «^  wliat  is  the  quotient  if  a^  is 
divided  by  a^  ?     What,  if  a^  is  divided  by  a"'  ? 

14.  How,  then,  is  the  exponent  of  a  quantity  in  the 
quotient  found  ? 

^  74.    Division  is  indicated  in  two  ways : 

1.  By  the  sign  -i-,  read  divided  by. 
Thus,  a  -^  b  shows  that  a  is  to  be  divided  by  b. 

2.  By  writing  the  dividend  above  the  divisor  with  a  line 
between  them. 

Thus,  -  shows  that  a  is  to  be  divided  by  b. 
b 

75.  Principles.  1.  TJie  sign  of  any  term  of  the  quotient 
is  -\-  when  the  dividend  and  divisor  have  like  signSj  and  — 
when  they  have  unlike  signs. 

2.  The  coefficient  of  the  quotient  is  equal  to  the  coefficient  of 
the  dividend  divided  by  that  of  the  divisor. 

3.  The  exponent  of  any  quantity  in  the  quotient  is  equql  to 
its  exponent  in  the  dividend  diminished  by  its  exponent  in  the 
divisor. 

76.  The  principle  relating  to  the  signs  in  division  may  be 
illustrated  as  follows : 


-f-ax  +  6  =  +  a6 
--ax  +  6  =  —  a6 
-{-ax  —  b  —  —  ab 
—  a  x  —  b  =  -\-ab. 


>.     Hence  < 


-\-ab-^-\-b  =  -\-a 

—  ab-7--\-b  =  — a 

—  ab-. —  6  =  +  a 
^-\-ab-{ —  b  =  —  a 


DIVISION.  65 


77.    To  divide  when  the  divisor  is  a  monomial. 

1.    Divide  Ux-yz^  by  —  7  xyz. 


—  Ixyz 


PROCESS.  Explanation.     Since  the  dividend  and  divisor 

.  .    2     „       \\2jwe  unlike  signs,  the  sign  of  the  quotient  is  — 
±i^'^       (Prin.  1.) 


—  2  XZ^  Then  14  divided  by  -  7  is  -  2  ;  x2  divided  hy 

X  is  X  (Prin.  3)  ;  y  divided  by  2/  is  1,  which  need 
not  appear  in  the  quotient ;  z^  divided  by  z  is  z^.  Therefore  the  quo- 
tient is  —  2  xz"^. 

2.   Divide  4  ax^y^  —  12  a^x^y^  —  20  a-xyh  by  2  axy, 

PROCESS. 


2axy 


4  gg^V  —  12  aVy^  —  20  a^xy^z 


2  x^y^  —  6  axy  —  10  ayh 


Explanation.  When  there  are  several  terms  in  the  dividend,  each 
term  must  be  divided  separately. 

Rule.  Divide  each  term  of  the  dividend  by  the  divisor  as 
follows  : 

Divide  the  coefficient  of  the  dividend  by  the  coefficient  of  the 
divisor.  To  this  quotient  annex  each  literal  factor  of  that 
term  of  the  dividend  with  an  exponent  equal  to  the  exponent  of 
that  letter  in  the  dividend  minus  its  exponent  in  the  divisor. 

Write  the  sign  -\-  before  each  term  of  the  quotient  ichen  the 
terms  of  both  dividend  and  divisor  have  like  signs,  and  — 
when  they  have  unlike  signs. 


3. 

4. 

5. 

6. 

7. 

8. 

ivide   10  a 

16  a; 

-14a6 

IBxf 

-  20  a'b' 

— 18  m-n 

y           6a 

2x 

lab 

-3xy 

-    4a6 

6  m 

Find  the  quotients  in  the  following : 
9.   30  a^bx"  -h  15  ax.  11.   21  ax'y  ^ -7  ay. 

10.    -  24  xYz^  -5-  8  ojV.  12.    -  9  abc^  -J-  -  3  abc. 


milne's  el.  op  alg. — 5. 


66  ELEMENTS  OF  ALGEBRA. 

13.  30nV^6n2.  18.  -  55  abc'd  ^  11  abc, 

14.  -12arV--12a;i/.  19.  27  a^z' -i- -  9  x'z. 

15.  -20oc^y*^-10y\  20.  -  120  m^n ---- 15  mn, 

16.  -  100  x'yz^  25  xyz.  21.  325  x^z' -i- 5  xyh\ 

17.  80  icV  -  20  xy^,  22.  -  65  a:^^  -^  -  13  icV. 

Divide  : 

23.  a^ici/  —  2  axy^  by  a^/. 

24.  9  xy  +  15  a;?/V  by  3  xyK 

25 .  14  a^&'^c  +  49  a'bc  by  7  abc, 

26.  —  a:^:^  —  3  a-^;  4-icV  by  —  xz. 

27.  4  c^d  -  14  cd-' by  2  cd. 

28.  —5x^y  +  10xy  —  15xy-hy5xy, 

29.  16  m-?i-  —  12  mhi  —  8  mn^  by  4  7/i?i. 

30.  15  ax^  —  25  6x-^?/  +  35  cxy^  by  —  5  a?. 

31.  9  x^yz  —  36  xyh^  -\-  45  ax^/a;^  by  9  xyz, 

32.  42a^-14a;2  +  28a;  +  35by  7. 

33 .  -  45  a2& V  _  60  a6c-  +  30  a& V  by  -  15  abc. 
'    34.  116  m' +  80  m*^  -  112  m^- 92  m  by  4  m. 

35.  3  x^yz'-  -  15  c^yh^  +  6  aj^"^  +  18  xYz  by  -  3  a^^/^^- 

36.  3  a.'S  -  6  o:^  +  9  o;^  -  12  x^  by  3  .^^. 

37.  30ary  +  60aj2/_45a.y_|.75a;by  15  a;. 

38.  24  abx  -  16  aby  +  32  a^bx^  -  8  a6  by  8  ab. 

39.  —x^y  —  x^yz  +  a;^'*^;  —  x^x^z^  -\-  xyh^  by  —  xy. 

40.  50  a^2/;33  _|.  35  ^2^2^  _  -1^5  ^^.2^^3  _  20  ft^^/V  by  5  xyz, 

41 .  2  ?i2ajy  -  3  7ia^2/'  -  ^  mnx^f  +  3  ri^ory  by  nx^y\ 

42.  a(a;  +  2/)^  -  «^(^  +  2/)^  +  a^b\x  +  3/)^  by  a(aj  +  yf. 


DivisioiNr.  67 

78.   To  divide  when  the  divisor  is  a  polynomial. 

1.    Divide  aF  —  3a-b  +  3  air  —  Irhy  a  —  b. 

PROCESS. 

a^  —     n^b 


a^-2ab  +  b^ 


-  2  a=^6  -h  3  aW 

-  2  a^6  +  2  a&' 

ab^  -  W 
ab'-  -  W 

Explanation.  The  divisor  is  written  at  the  right  of  the  dividend, 
and  the  quotient  below  the  divisor.  The  first  term  of  the  divisor  is 
contained  in  the  first  term  of  the  dividend  oP'  times.  Therefore  d^  is 
the  first  term  of  the  quotient,  o?-  times  tlie  divisor,  a  —  6,  is  a^  —  a%, 
and  this  subtracted  from  the  partial  dividend  leaves  a  remainder  of 
—  2  a^hy  to  which  the  next  term  of  the  dividend  is  annexed  for  a  new 
dividend. 

The  first  term  of  the  divisor  is  contained  in  the  first  term  of  the  new 
dividend  —2  ah  times,  consequently  —  2  a?>  is  the  second  term  of  the 
quotient.  —2  ah  times  the  divisor,  a  —  6,  is  —  2  a%  +  2  aft^,  and  this 
subtracted  from  the  second  partial  dividend  leaves  a  remainder  of  ah'^^ 
to  which  the  next  term  of  the  dividend  is  annexed  for  a  new  dividend. 

The  first  term  of  the  divisor  is  contained  in  the  first  term  of  the 
new  dividend  6^  times,  hence  6'^  is  the  third  term  of  the  quotient. 
&2  times  a  —  6,  is  ah"^  —  h^,  which  subtracted  from  the  third  partial 
dividend  leaves  no  remainder.     Hence  the  quotient  is  a"^  —  2  ah  -\-  h^. 

Rule.  Write  the  divisor  at  the  right  of  the  dividend, 
arranging  the  terms  of  each  according  to  the  ascending  or 
descending  powers  of  one  of  the  literal  quantities. 

Divide  the  first  term  of  the  dividend  by  the  first  term  of  the 
divisor,  and  write  the  result  for  the  first  term  of  the  quotient. 

Multiply  the  divisor  by  this  term  of  the  quotient,  subtract 
the  product  from  the  dividend,  and  to  the  remainder  annex 
one  or  more  terms  for  a  new  dividend. 

Divide  the  new  dividend  as  before,  and  continue  to  divide 
until  there  is  no  remainder,  or  until  the  first  term  of  the  divisor 
is  not  contained  in  the  first  term  of  the  dividend. 


68 


ELEMENTS  OF   ALGEBRA. 


If  there  is  a  remainder  after  the  last  division,  write  it  over  the 
divisor  in  the  form  of  a  fraction,  and  annex  it  with  its  proper  sign  to 
the  part  of  the  quotient  previously  obtained. 


15x2-    sxy  -  12  y2  i  Sx-\-2y 

16x^+  10 xy  \  6x-6y 

-  ISxy-  12  2/2 

-  ISxy  -  12?/2 


Divide  : 


2c-^d 


4c-3cZ  + 


8c2-26c(iJ+  15f72  +  8 

8c^-20cd 

-6cd-\-  l^d^ 
-6cd  + 15(^2 

+  8 

4. 

«2  -f  a6  —  ac  —  be  \  a  +  b 
a^  +  ab  \  a  —  c 


8 


2c-bd 


—  ac 

-bc 

-  ac 

-be 
5. 

x^- 

-y' 

x-y 

x^- 

-  xh/ 

x^  +  xy  +  ?/2 

x'y  - 

_  yS 

x^y- 

-  xy'^ 

xy^- 

2,3 

xy'2- 

_y!_ 

6.  x^-{-2xy  +  y^hjx-}-y. 

7.  m^  —  n^  hj  m  ^  n. 

8.  oi:^-^3x^y  +  3xy'^-\-y^hyx  +  y, 

9.  x'^Sx-lShjx-^e. 

10.  ci^  +  Sx-i-Why  x-{-3, 

11.  d'-10ab-24.b^hy  a-{-2b. 


DIVISION.  69 

12.  ic2_4^^4  by  X  — 2. 

13.  a;^—  49  by  aj2_j_7. 

14.  a*^-16by  a'^-4. 

15.  10iK'-  +  14a?-12by  2aj  +  4. 

16.  2i^-a;'-  +  3aj-9by  2x-3. 

17.  16a'-24.ab-{-9b^hj  4:a-3b. 

18.  a-^  +  b''  by  a  +  &. 

19.  x'-Shj  x-2. 

20.  a^ -\- a^y  +  ay^-{-y*  hy  a -{-y. 

21.  aj3_5a;2-a?  +  14bya?--3a;-7. 

22.  ar^-86aj-140by  a;-10. 

23.  6x^  +  4.f-^6xy'^-16x^yhj3ccF-y^-2xy. 

24.  18a;-l-17a^by  1-07. 

25.  a'-\-4.aV-{-16x'hy  a'-{-2ax-{-4:X^. 

26.  27aj3  +  8/by  3iB-f-22/. 

27.  4aj^-.r^4-4a;by  2  +  2aj2  +  3a;. 

28.  x^  —  ^x^  —  exy  —  y^hyx^-^-Sx-^-y. 

29.  36  +  a;^-13a^by  e  +  aj^  +  Sa;. 

30.  1  — a;— 3ic''^  — a^  by  l4-2a;  +  a;2. 

PROBI.EMS. 

79.    1.   Express  the  sum  of  x  and  y  divided  by  2. 

2.  If  a;  oranges  are  worth  10  cents,  how  much  is  one 
orange  worth  ? 

3.  How  many  apples  at  2  cents  apiece,  can  be  bought  for 
m  cents  ? 

4.  Express  the  quotient  of  b  divided  by  a. 

5.  If  5  bananas  cost  c  cents,  what  will  2  bananas  cost  ? 


25 


70  ELEMENTS  OF  ALGEBRA. 

6.  How  many  days  will  it  take  a  man  to  earn  $  15,  if  he 
receives  a  dollars  a  day  ? 

7.  A  farmer  has  a  cows,  ^  as  many  horses,  and  three 
times  as  many  sheep  as  horses.  How  many  animals  has  he 
altogether  ? 

8.  Henry  and  James  together  had  d  marbles ;  Henry 
had  c  times  as  many  as  James.  What  represents  the  num- 
ber that  each  had  ? 

9.'  What  is  the  cost  per  barrel,  if  b  barrels  of  flour  cost 
the  sum  of  m  and  n  dollars  ? 

10.  A,  B,  and  C  start  in  business.  A  furnishes  2a 
dollars,  which  is  c  times  as  much  as  B  furnishes;  and  C 
furnishes  b  times  as  much  as  B.  How  much  do  B  and  C 
furnish  ? 

11.  A  grocer  mixed  together  equal  quantities  of  three- 
kinds  of  colfee,  worth  a  cents,  b  cents,  and  c  cents  per 
pound.     What  is  the  cost  of  one  pound  of  the  mixture  ? 

GENERAL    REVIEW   EXERCISES. 

80.  1.  Add  6{x-\-7j)^3z-S,  2{x  +  y)-2z-{-^,  Sz 
-S(x  +  y), 

2.   Add  a'+b'-\-4:ab,  5a^-5b--5ab,  -3a'-{-2b'-4.ab. 

S,    Add  ^x  +  ^y  +  z,  x-j-iy-\-^z,  ^x-^y-^^z. 

4.  Add  2x'^—5xy''-{-6,  Sxy''+4:y''—5x'^,  ex^'—lO  +  Txy^ 

5.  Add  f  (m  — n)4-|(m -f  n),  f  (m  —  n)  —  f  (m +  n), 
^(m  —  n)  —  f(m  +  n), 

6.  From  9a;  +  9a;--3a^  subtract  2 a;  -  2 a;^  -  12 aj^. 

7.  From  fa  — ^6  +  ^c  subtract  ^a  +  ift  — fc. 

8.  From  Bx'^y —  Tx'^y^ -{-Wx'^z''  subtract  2x^y^  —  3x"'y 
-f-5a;"*2;^ 


GENERAL  REVIEW  EXERCISES.  71 

9.  From  S(x+yy-2z^+S  subtract  2z^-x^-\-4:{x-\-yy. 

10.  From  Sa^  —  2a^x  —  7  subtract  7  +  3a^ —  a^x-\-2m. 

11.  Multiply  m^  — m^  +  m^  — m  +  1  by  m  +  1. 

12.  Multiply  a  +  b  +  c-j-e  by  a -{- b -{- c  -{- e. 

13.  Multiply  x^'{-2x^y-\-4.xY  +  Sxf-\-16y^  by  x^2y. 

14.  Find  the  product  of  a;  —  10,  x -\-  4,  and  aj  +  6. 

15.  Find  the  product  of  a  —  b,  a  +  &,  and  a^  —  b\ 

16.  Find  the  product  of  2a— 5,    2a  — 5,   2a  — 5,    and 
2a +  5. 

17.  Divide  4:a' -  5a^b^ -\-b^  by  2a2-3a6  +  62. 

18.  Divide  a^  +  1  by  a  +  1. 

19.  Divide  ni^  —  c^  -\- 2  cz  —  z^  by  m  +  c  —  z. 

20.  Divide  l  —  x  —  Sx'^—x^  by  l  +  2a;4-a^. 

21.  Divide  6x'-96  by  3a; -6. 
Find  the  value  of  x  in  the  following : 

22.  10a;-3a;4-4  =  .T  +  10-2aj-h8  +  a?. 

23.  6a; -  13  -  9 a;  +  a;  =  4a;— 12  +  3 a;  — 6aj- 13. 

24.  5(a;  +  l)  +  6(a;  +  2)=6(a;  +  7). 

25.  3(a;  +  l)  +  4(a;  +  2)  =  6(a;  +  3). 

26.  ia;-4  +  fa;=  16  +  ia;-10. 

27.  a;(a;4-5)-6  =  a;(a;-l)  +  12. 

Find  the  values  of  x  and  y  in  the  following : 

J  3a; -72/ =  13  I  cSx-15y  =  19^ 

^^'     (43; +  22/ =  40 1  ^^'     |2a;+      y  =  19\ 

^  3a; -32/  =  -    6  I  r4a;+    22/  =  24| 

^^'     \7y^3x=     22)  ^^*     (3a;-      y=    8) 


72  ELEMENTS  OF   ALGEBRA. 

32.  A  man  being  asked  how  much  money  he  had,  replied 
that  $  25  more  than  three  times  what  he  had  would  equal 
f  775.     How  much  money  had  he  ? 

33.  A  man  drove  155  miles  in  three  days.  On  the  second 
day  he  drove  15  miles  farther  than  on  the  first,  a.nd  on  the 
third  day  he  drove  20  miles  farther  than  on  the  first.  How 
far  did  he  drive  each  day  ? 

34.  A  and  B  start  from  two  towns  231  miles  apart  and 
travel  toward  each  other.  A  goes  15  miles  per  day,  and  B 
goes  18  miles  per  day.     In  how  many  days  will  they  meet  ? 

35.  A  man  worked  20  days  during  a  certain  month.  A 
part  of  the  time  he  received  $  1  per  day,  and  part  of  the 
time  $  1^  per  day.  If  he  received  $  25  for  his  wages,  how 
many  days  did  he  work  at  f  1,  how  many  at  $  1^  per  day  ? 

36.  David  and  his  father  earned  $  100  during  a  certain 
month.  David  earned  $  10  more  than  ^  as  much  as  his 
father.     How  much  did  each  earn  ? 

37.  A  man  bought  an  overcoat,  a  suit  of  clothes,  and  a 
pair  of  boots.  The  overcoat  and  the  suit  of  clothes  cost 
$  55 ;  the  overcoat  and  the  boots  cost  $  30 ;  and  the  suit  of 
clothes  and  the  boots  cost  f  35.  What  was  the  cost  of 
each? 

38.  A  and  B  each  own  half  of  a  flock  of  120  sheep.  In 
settling  the  partnership,  A  takes  45  sheep,  B  takes  75  sheep 
and  gives  A  $  45  to  make  the  division  equal.  What  is  the 
value  of  one  sheep  ? 

39.  A  merchant  has  one  quality  of  sugar  worth  5  cents 
and  another  worth  8  cents  a  pound.  He  wishes  to  make  a 
mixture  of  100  pounds  worth  7  cents  a  pound.  How  many 
pounds  of  each  quality  must  he  take  ? 


FACTORING. 


81.  The  quantities  which,  when  multiplied  together,  pro- 
duce a  quantity,  are  called  Factors  of  the  quantity. 

Thus,  «,  b,  and  (x  -f  p)  are  the  factors  of  ah(x  +  y). 
a,  6,  and  (x  +  y)  are  called  the  prime  factors  of  ab{x  +  2/),  because 
they  have  no  common  factors  besides  themselves  and  1. 

82.  A  factor  of  a  quantity  is  an  Exact  Divisor  of  it. 

83.  The  process  of  separating  a  quantity  into  its  factors 
is  called  Factoring. 

84.  To  factor  a  polynomial  when  all  the  terms  have  a  com- 
mon factor. 

1.   What  are  the  factors  of  4  a^m  —  6  am^x  -f  10  aVx^  ? 


PROCESS. 


2  am 


4  a^m  —  6  am^x  +  10  a^mV 


2a  —  3 mx  -{-  5 amx^ 


Explanation.  By  examining  the  terms  of  a  polynomial,  we  find 
that  2  am  is  the  highest  factor  common  to  all  the  terms.  Dividing  by 
2  am,  we  obtain  the  other  factor.  Hence,  the  two  factors  are  2  am 
and  2a  —  S mx  +  5 amx^. 

The  same  result  may  be  secured  by  separating  the  terms  of  the 
polynomial  into  their  prime  factors  and  then  selecting  the  common 
factors. 

Rule.  Divide  the  polynomial  by  the  highest  factor  or  divisor 
common  to  all  the  terms.  The  divisor  and  quotient  will  be  the 
fOjCtors  sought. 

73 


74  ELEMENTS  OF  ALGEBRA. 

Find  the  factors  of  the  following  polynomials  : 

2.  lSx^-27xy.  12.  20  m^  -  50  m^  +  30  m*. 

3.  15a^2/2  +  20xY  13.  xy' -\- 9  xY -]- 27  a^f, 

4.  12  mV  —  48  mn^.  14.  5  am^  +  10  mn  +  15  mn^. 

5.  5a6c-5ac2  4-15a6-c.  15.  4.5  xYz  +  60  x'yz^ 

6.  6aj2  +  4a;?/-8a^.  16.  39 a^y -  65  a^i/ +  ^1  ^2A^- 

7.  9 x'y^- 6 x^y-\- 12 a^yz.  17.  32 a'^6«  +  96 a^6«  -  8 aW. 

8.  6  a^b  +  21  a^b- IS a%^  18.  25  a?i  +  75  am^  -  15  anl 

9.  Sa'b-abc-abd.  19.  15  aar^  -  35  ory  -  55  fta^^ 

10.  6a*6'  +  36a'62-42a%l    20.    45?>-V  +  60^^  -  30aj%. 

11.  15a26^+20aV-25a^6l     21.   72aY-36ay-108a2y\ 

85.   To  factor  a  polynomial  when  some  of  the  terms  have  a 
common  factor. 

1 .  Factor  S  xy  -\-  3  xz  -{-  ay  -{- az. 
Solution.  S  xy  -\-  S  xz  -{-  ay  -\-  az 

=  'Sx(y  +  2)-\-  a(y  +  z) 
=  (Sx  +  a)(y  +  z) 

^  Factor  the  following : 

2.  ax -\- by  -\- bx -{-  ay.  10.    2  a*  —  a*^  +  4  a  —  2. 

3.  a6  +  26+3a  +  6.  11.    ax  —  ab  —  bx -}- b\ 

4.  9  +  Sx  +  3y-i-xy.  12.    x^ -\- x'^ -{- ax -\- a. 

5.  ax-{-ay—bx  —  by,  13.    ax'^ -{- ay^  —  bx^  ^  by\ 

6.  xy -\- 6  —  bxy  —  6b.  14.    1  +  a^  — a"  — a^ 

7.  y^  —  y^-\-y  —  l.  15.    1  —  a?  —  .r^  4- a^. 

8.  x^ -\- x^y  —  xy^  —  "(f.  16.    x-y  —  x-z  —  xy'^ -{- xyz. 

9.  a;^^^  +  6a;^  —  6y  —  6^.  17.    a -{- ay  -{-  ay'^  +  a^/^. 


FACTORING.  75 

86.  To  separate  a  trinomial  into  two  equal  factors. 

(m-^n)(m  -{-  n)=  m^  +  2  mn  +  nl 
(m  —  n)(m  —  n)^7n^  —  2  mn  +  ri^. 

1.  What  is  the  product  of  (7n -{- n)  (m -\- n)  ?  What, 
then,  are  the  factors  of  m^  +  2  mn  +  n^?  How  are  the 
terms  of  the  factors  found  from  the  trinomial  ? 

2.  What  is  the  product  of  (m  —  n)(m^n)?  What, 
then,  are  the  factors  of  m^  —  2  mn  +  n^  ?  How  are  the 
terms  of  the  factors  found  from  the  trinomial  ? 

3.  What  term  of  the  trinomial  determines  the  sign  which 
connects  the  terms  of  the  binomial  factors  ? 

87.  One  of  the  two  equal  factors  of  a  quantity  is  called 
its  square  root. 

Rule.     Find  the  square  roots  of  the  terms  that  are  squares 

and  connect  these  roots  by  the  sign  of  the  other  term.     The 

result  will  be  one  of  the  equal  factors. 

The  other  term  must  always  be  twice  the  product  of  the  square 
roots  of  the  terms  that  are  squares. 

Find  the  equal  factors  of  the  following  trinomials  : 

1.  x'^-{-2xy-{-y^  9.  4:X^  -{-Sxy -{- 4:y\ 

2.  x^-i-Ax  +  A.  10.  x^-lOx-^  25. 

3.  4:a"  —  4:ab-{-b\  11.  l-h2z-\-z\ 

4.  9m^  +  6mn-\-n\  12.  a^ -\- 4:  a^b^ -\- 4:b\ 

5.  ar-4.xy-]-4:y\  13.  x"" -  G xz -]- 9 z\ 

6.  2/'  +  22/  +  l.  14.  aj2^20aj  +  100. 

7.  a-b^—Sab-\-16.  15.  m^  +  8 m?i  +  16 n^. 

8.  4n2-20n  +  25.  16.  9x^-18xy  +  9y\ 


76  ELEMENTS   OF   ALGEBRA. 

17.  9  4-6a2-fa^  23.  a^  +  18 a^ -f- 81. 

18.  Aa'x'-Aa'b^x-hb',  24.  100 aj^  _  20  a^  +  1. 

19 .  aW  —  8  amn  +  16  nl  25 .  m V  +  4  mn  +  4. 

20.  25a;*  +  80aj2+64.  26.  x^""  +  2 x'^y'' -{- y^\ 

21.  a*  +  4a2iK2  +  4aj^  27.  16  4-16m2  +  4m*. 

22.  46V-12&cd4-9d2^  28.  1  -  8 a?"*  +  16 a;^^ 

88.   To  resolve  a  binomial  into  two  binomial  factors. 

(^a+b)(a-b)  =  a^-b'. 

1.  What  is  the  product  of  (a  -{-b)(a  —  b)?  What,  then, 
are  the  factors  of  a^  —b^?  How  do  these  two  factors 
differ  ? 

2.  What  is  the  product  of  {x  +  3)  (a;  -  3)?  What,  then, 
are  the  factors  of  a?^  —  9  ? 

EuLE.  Find  the  square  root  of  each  term  of  the  binomial  and 
make  the  sum  of  these  square  roots  one  factor  and  their  dif- 
ference the  other. 

A  binomial  cannot  be  factored  by  the  above  rule  unless  the  second 
term  is  negative  and  the  indices  of  the  powers  are  even  numbers. 

Sometimes  the  factors  of  such  a  quantity  may  themselves  be  re- 
solved into  factors. 


Thus, 

x^-y^ 

=  (x2  +  2/2)(X-^-2/2) 

Factor  the 

following : 

1.    x'-y^ 

6.    aj2-25. 

2.    a;^  — 4. 

7.    121  a.-^- 1002/'. 

3.    a- -9 62. 

8.    16a- -96-. 

4.    m'-l. 

9.    a;^-144. 

5.   a'b'--c\ 

10.    l-z\ 

FACTORING.  77 

11.  x'^  —  y^  21.  m^  —  ^2^. 

12.  a;- -16.  22.  225 or  -  100 2/'. 

13.  9a;2-36.  23.  162/^-256. 

14.  Ax' -25,  24.  aV-Odl 

15.  a;^-81.  25.  a;- -121. 

16.  mV~  640^2  23  c'd' -  1. 

17.  4a2-166l  27.  25m2- 225711 

18.  x^-625.  28.  x^-1, 

19.  9x^  —  81.  29.  a;2m__^2« 

20.  25a-62_49c^d^  30.  a:«- -  169. 

89.  To  factor  a  quadratic  trinomial. 

(x-\-3){x  +  2)=:x^  +  5x-\-6, 
(x-\-3){x-2)  =  x^-\-x-6, 
(x-3){x-{-2)  =  a^-x-6. 
(^x-3)(x-2)  =  x''-5x-{-6. 

1.  In  the  above  examples,  what  terms  of  the  product  are 
alike  ? 

2.  How  may  the  first  term  of  each  factor  be  found  from 
the  product  ?  How  may  the  second  term  of  each  factor  be 
found  from  the  product  ? 

3.  How  is  the  coefficient  of  the  second  term  in  the 
products  found  from  the  last  terms  of  the  factors  ?  How, 
then,  may  the  sign  of  the  second  term  of  each  factor  be 
found  from  the  second  term  of  the  product  ? 

90.  A  trinomial  in  the  form,  x^  ±  ax  ±  b,  in  which  b  is 
the  product  of  two  quantities,  and  a  their  algebraic  sum, 
is  called  a  Quadratic  Trinomial. 


78  ELEMENTS  OF   ALGEBRA. 

1.  Eesolve  a;^  —  9aj  + 18  into  two  factors. 

PROCESS. 

2      Q      I   1 Q  Explanation.     The  first  term  of  each  factor  is 

aj-  —  y  aj  +  15  evidently  x.     Since  18  is  the  product  of  the  other 

r  18  X  1  two  quantities,  and  —  9  is  their  sum,  we  must 

j^g  _  <{      9x2  select  from  the  different  pairs  of  factors  of  18  those 

I      fi  V  S  ^^^  whose  sum  is  —  9.    Therefore,  —  6  and  —  3 

^  are  the  other  terms.     Hence,  x  —  6  and  x  —  3  are 

—  ^d^  —  ^—o  the  factors  required. 

(a7-6)(a;-3) 

2.  Factor  a;^ -f- 4  a;  -  21. 

Solution.     21  =  3  x  7,  or  21  X  1. 

The  factors  whose  algebraic  sum  is  +  4  are  +  7  and  —  3. 

. '.  (x  +  7)  and  {x  -  3)  are  the  factors. 

Factor  the  following : 

3.  a^+Ca+S.  17.  aj^ -  10 a? - 200. 

4.  x2  +  5a;  — 14.  18.  a2-6aaj  — 55a;^. 

5.  i^2_0^_7  19.  aj2-14a;  +  40. 

6.  m--10m+9.  20.  x^+xy-20y\ 

7.  y'-^^h-\-Q^,  21.  7n2  +  10m  +  24. 

8.  n^-n-2.  22.  a'-A.ah-2lh\ 

9.  n''  +  n-2.  23.  a^- 14a; +  48. 

10.  a.^-7a;-18.  24.  x'^lx-lS. 

11.  aj2  +  4a;-12.  25.  0^  +  0-420. 

12.  a' -5a -24:.  26.  a^  + lla&  +  2862.     . 

13.  x^-{-5x-S6.  27.  a:2n_9^n_^20. 

14.  y''-Syz-^2z\  28.  aj^  _  3 a^^  -  154. 

15.  m^-6mn-16n^  29.  m'  +  2m-255. 

16.  a'  +  dax  +  Sx^  30.  a^^  +  12 x'^  +  35. 


FACTORING.  79 

91.  To  factor  the  sum  or  the  difference  of  two  cubes. 

(a3  4-  b')  _j-  (a  +  5)  =  a^  -  a6  +  b^ 

92.  Principle.  The  sum  of  the  cubes  of  two  quantities 
is  divisible  by  the  sum  of  those  quantities,  and  the  difference 
of  the  cubes  of  two  quantities  is  divisible  by  the  difference  of 
those  quantities. 

1 .  Factor  a^  —  oc^. 

Solution.        d^  —  x^  is  divisible  hy  a  —  x  (Prin.) 
(^3  _  x^)  ^(a-x)=  a^  +  ax  +  x2 

.-.  a^-x^={a- ic)(a2  +  ax  +  x^) 

2.  Factor  7^ -|- 5^. 

Solution.        r^  4-  s^  is  divisible  by  ?•  +  s  (Prin.) 

(r3  +  s^)  -t-  (r  +  s)  =  r2  -  rs  +  s2 

.-.  r^  +  s^={r  +  s)(r2  -  rs  4-  s^) 

3.  Factor  a^  — 6 V. 

Solution.        a^  —  hH^  is  divisible  by  a  —  6c 

(^3  _  ;^3c3)  -^  («  _  he)  =  a2  +  abc  +  62c2 

.-.  a^  -  6'^c^  =  (a  -  6c)  (a^  +  abc  +  b'^c^) 

Observe  carefully,  in  the  examples  solved  above,  the 
quantities  in  the  quotients  and  the  signs,  and  you  will  be 
able  to  write  out  the  factors. 

Factor  the  following : 

4.  x^-{-y\  10.  s^-{-t\  16.  x'-^fz'. 

5.  a^+fe^.  11.  c^-d^  17.  a^  +  ftV. 

6.  m^  —  n^.  12.  2/^  +  1.  18.  mV  +  c^d^. 

7.  a^-ft^  13.  a^-1.  19.  aV -Wy\ 

8.  m^  +  n^  14.  a^  — cW.  20.  aj^  +  l. 

9.  Q^  —  f,  15.  m^H-sl  21.  2/^  —  1. 


80  ELEMENTS   OF   ALGEBRA.  \ 

\ 

EQUATIONS  SOLVED  BY  FACTORING.  ^ 

j 

93.     1.    Find  the  value  of  x  in  tlie  equation  x--\-l  =  o,      \ 

PROCESS.  Explanation.      Transposing    the  = 

2       -,       ^  known  quantity  to  the  second  mem- ; 

^  +-•-  =  *>  ^gj.^   the  first  member  contains  the  ■ 

ar  =  5  —  1  second  power,  only,  of  the  unknown  i 

X- =  4,  quantity.     Separating  eacli  member  j 

^    ^^      r>    o  _,.       o         o     into  two  equal  factors,  the  equation' 

becomes  ic-a;  =  2.2    or    — 2-—  2.1 
.'.  x=  ±  Z  Since  each  member  is  composed  of  j 

two  equal  factors,  a  factor  in  each: 
must  be  equal.     Hence,  x  =  +  2  or  —  2  ;  or  x  =  i  2. 

The  sign,  ± ,  called  the  Ambiguous  Sign,  is  a  combination  \ 

of  the  signs  of  addition  and  subtraction.  \ 

Thus,  a  it  6  indicates  that  b  may  be  added  to  or  subtracted  from  a,  \ 

2.    Find  the  value  of  x  in  the  equation  a;^  +  5  =  30  -f  11.    j 
Solution.  x^  +  5  =  30  4-  H  •     • 


x2  =  30  +  11  - 

5 

x2  =  36 

X'X  =  6'Q 

or  — 

6.-6 

.'.  x=  ±Q 

Fin 

d  the  value  of 

X  in  the  following  equations  : 

3. 

a:2_4  =  5. 

12. 

a;2  +  21  =  25. 

4. 

a;2-9  =  16. 

13. 

aj2  +  l  =  82. 

6. 

X' -25  =  24.. 

14. 

a;2  +  12  =  48. 

6. 

aj2  _  1  =  3. 

15. 

x'-15  =  10. 

7. 

0^2  +  4  =  20. 

16. 

aj2  _  10  =  90. 

8. 

a;2  +  5=41. 

17. 

a^  +  15  _  8  =  8. 

9. 

a;2  +  2  =  ll. 

18. 

a;2+3-6=5-24-10 

10. 

aj2-8  =  8. 

19. 

x'-4.a'  =  21a\ 

11. 

«2  +  7  =  ll. 

20. 

a^  +  3c2=7cl 

FACTORmG.  81 

94.    21.  Findthe  value  of  a;  in  the  equation  aj^ -1-4  a; -f  4=16. 

PROCESS.  Explanation.         Since 

2       .    ^        .       ^  ^  each  member  of  the  equa- 

00  -h  4  a;  -h  4  =  lb  ^^^^  ^^^  1^^  resolved  into 

(a;  +  2)(a;-f2)  =  4-4or  — 4.  — 4  tico  equal  factors,  one  fac- 

.*.  a; -f  2  =  4  or  —4  ^^^  of    ^^^^    first    member 

J  OP  must  be  equal  to  one  fac- 

and  a;  =  2  or  —  b  ,       .  ..  a  ^ 

tor  of  the  second  member. 

Hence,   x  +  2  =  4   or  —  4, 
and  X  is  found  to  have  two  values,  2  or  —  6. 

22.  Find  the  value  of  x  in  the  equation  a^  -f  6  a;  +  9  =  49. 
Solution.  x^  +  6  ic  +  9  =  49 

(x  +  3)(x  +  3)=7.7or-7.-7 

.-.  cc  +  3  =  7  or  -  7 

and  X  =  4  or  —  10 

Observe  that  the  first  member  cf^nnot  be  separated  into  two  equal 
factors  except  when  the  trinomial  is  a  perfect  square  (Art.  87). 

Find  the  values  of  the  unknown  quantities  in  the  follow- 
ing equations  : 

23.  /- 10  ?/-h  25  =  16.  32.  a?- -f  24 a; -f  144  =  225. 

24.  a;2-8a;  +  16  =  81.  33.  a;^ -f- 2  a; -f- 1  =  36. 

25.  a;--16a;-f  64  =  9.  34.  y'^ -Uy -\- A9  =  9. 

26.  z'-{-12z  +  36  =  64..  35.  z' +  30 z -\- 225  =  625, 

27.  a;2 -f  14  a;  +  49  =  100.  36.  ar' -^  24a;-f  144  =  169. 

28.  a:^  _  20  a; -f  100  =  25.  37.  a^ -f  40  a; -f  400  =  900. 

29.  2/' -18  2/ -f  81  =  16.  38.  a;^  -  26a; -j- 169  =  196. 

30.  x'-{-22x-^121=:lU.  39.  y'-2y-\-l  =  25. 

31.  a;2-f  6  a; -f- 9  =  4.  40.  a;--f  50a; -f- 625 =1600. 

MILNE ^S    EL.    OF    ALG.  — 6. 


82  ELEMENTS  OF   ALGEBRA. 

95.  41.    Find  the  value  of  x  in  the  equation  a^-f  4  a;  =  —4. 

PROCESS.  Explanation.     By  transposing  —  4  to 

the  first  member,  that  member  becomes  a 

X  -f-  4  X'  =  —  4  perfect  square  which  may  be  resolved  into 

aj2_|_4/p_|_4_Q  lyQQ  equal  factors,  {x  +  2)  (x  +  2).    Since 

( X  A-  2^  (x  A-  2\  =^  Q  the  product  of  these  factors  is  0,  one  of 

on  ^^®  factors  must  be   0;   and  since  both 

.*.   i»  +  ^  =  li  factors  are  the  same,  each  factor  is  equal 

and                    x=  —2  to  0.     Hence,  the  value  of  x  is  -  2. 

42.    Find  the  value  of  x  in  the  equation  a;-  —  8  a;  =  —  16. 

Solution.  ic2  _  g  ^j  ^  16  _  0 

(x  -  4)  (x  -  4)  =  0 

...  x-4  =0 

and  x  =  4 

Find  the  values  of  x  in  the  following  equations : 

43.  a;2 -f.l0a;4-25  =  0.  52.  aj- +  20  a;  4- 100  =  0. 

44.  a;2  4-12a;  +  36  =  0.  53.  aj- +  24  a;  =  -  144. 

45.  a;2  +  6a;  4-0  =  0.  54.  a;- -f- 36  a;  =  —  324. 

46.  a;2-18aj  +  81  =0.  55.  a;^  -  50  a?  =  -  625. 

47.  aj2-f  16  a; -f- 64  =  0.  56.  a^  +  22  a;  =  -  121. 

48.  aj2 -I- 26  .T  +  169  =  0.  57.  a;^  -  100  a;  =  -  2500. 

49.  a;--14aj  +  49  =  0.  58.  a;- +  44  a;  =  -  484. 

50.  a;2  +  30a;  +  225  =  0.  59.  a;^  +  60  a;  =  -  900. 

51.  x^ -28  aj  + 196  =  0.  60.  x' +  80  a;  =  -  1600. 

96.  61.    Find  the  values  of  x  in  the  equation  x^-\-^^ 
- 14  =  0. 

PROCESS.  Explanation.    Factoring  the  first 

member  of  the  equation,  the  factors 

a^  +  5  a:  —  14  =  0  are  (x  +  7)  (:«  -  2) .     Since  the  prod- 

(^x  ^l)(x  —  2)  =  0  uct  of  the  factors  is  equal  to  0,  one  of 

•    a;4-7  =  0oraj-— 2  =  0  ^^®  factors  is  equal  to  0.     Solving, 

*  *                      '      ^                 ^  the  values  of  x  are  found  to  be  —  7 

and         a;  =  —  7,  or  a;  =  2  ^p  2. 


FACTORIlSrG.  83 

62.    Find  the  values  of  x  in  the  equation  cc^  -j-  a;  —  72  =  0. 
Solution.  ic^  +  oj  -  72  =  0 

.-.  cc  +  9  =  0,  orx-8=0 
and  X  =  —  9,  OY  X  =  8 

Find  the  vakies  of  x  in  the  following  equations : 

63.  x'^—2x  —  15  =  0.  72.  a;- +  15  oj  4-50  =  0. 

G4:.  x^-{-6x-{-5  =  0.  73.  x^-x-2=z0. 

65.  a;- +  10  a; +9  =  0.  74.  o?'^- 18  a;  + 77  =  0. 

66.  i»2-8a^  +  16  =  0.  75.  a?- +  2  a.' -  120  =  0. 

67.  a;2+2ar-48  =  0.  76.  ar'- 22  a?  -  75  =  0. 

68.  a?- +  13  a; +  40  =  0.  77.  x"  -  6ax -^8a^  =  0. 

69.  a:^- 5  a? -24  =  0.  78.  a.-^  +  3a;  -  54  =  0. 

70.  a;- +  7  a; +  12  =  0.  79.  a.-^  -  4  aa;  —  96  a- =  0. 

71.  a;2+9a;-22  =  0.  80.  a^  +  11  6x  +  246- =  0. 

97.    81.    Find  the  value  of  x  in  the   equation  aj^  +  Ga; 

+  7  =  23. 

PROCESS. 

a^^    6a;+  7  =  23 

x^+6-x+    7+2  =  23  +  2 
aP-\-    6.^;+  9  =  25 
(a; +  3)  (a; +  3)  =  5- 5  or  -5.-5 
.-.  aj  +  3=5or— 5 
and  a;  =  2  or  —  8 

Explanation.  Equations  like  this  may  be  solved  in  the  same 
manner  as  the  equations  immediately  preceding,  by  transposing  all  the 
quantities  to  the  first  member  and  then  factoring ;  or  they  may  be 
solved  by  making  the  first  member  a  perfect  square  by  adding  to  or 
subtracting  from  both  members  some  number.    The  first  member  of 


84  ELEMENTS  OF  ALGEBRA.  i 

the  equation  is  a  trinomial.     A  trinomial  is  a  perfect  square  when  \ 

it  is  composed  of  two  terms  that  are  perfect  squares  and  when  the  ; 

other  term  is  twice  the  product  of  the  square  roots  of  the  terms  that  ^ 

are  squares,     x'^  -\-  6x  are  two  terms  of  the  trinomial  which  is  to  be  | 
made  a  square,  but  the  third  term  is  to  be  found.     Since  the  second 

term,  6  x,  is  twice  the  product  of  the  square  roots  of  the  terms  that  are  ] 

squares,  and  the  square  root  of  one  of  the  terms  is  x,  if  6  x  is  divided  ; 

by  2  Xy  the  square  root  of  the  other  term  that  is  a  square  will  be  found.  I 

Dividing,  the  quotient  is  3,  and  S^,  or  9,  is  the  third  term  of  the  tri-  1 

nomial.     Since  the  given  term  is  7,  2  must  be  added  to  both  mem-  i 

bers  to  make  the  first  member  a  perfect  square,  giving  x"^-f  6x-f9=:25.  : 

Factoring,     (x  +  3)  (x  +  3)  =  5  •  5   or    —  5  •  -  5.     Whence,   x  =  2  or  ; 

-8.  i 

82.  Find  the  value  of  x  in  the  equation  aj^~-12a;-f-33=46.  ] 
Solution.           x2  -  12  x  +  33  =  46                           •  ; 

x2  -  12  X  +  33      +    3  =  46  +  3  \ 

x2  -  12  X  -f  36  =  49  I 

(x-6)(x-  6)=7.7or -7. -7  .  ; 

.-.  X-    6=    7  or     -7  I 

and  X  =  13  or     —  1  \ 

Solve  the  following  equations  : 

83.  a;2  +  10a;  +  20  =  lL  92.    a:- - 24 a; -f  122  =  3.  ^ 

84.  i»2^8x  +  12  =  32.  93.   a;^  -  30  a; -j- 220  =  76.  ] 

85.  a;2_18a;+80  =  15.  94.    aj^  ^  40 a;  +  200  =  425.  \ 

86.  aj2  +  4a;-f  2  =  7.  95.    a;^  _8a;  + 15  =  99.  ] 

87.  a^-20a;  +  85  =  10.  96.    aj^  +  12 aj  +  27  =  40.  j 

88.  a^  +  14a;-f  45  =  60.  97.    a;^- 38x  + 360  =  8.  ] 

89.  a;2  +  22 a;  +  100  =  60.  98.    y^  +  2y-l  =  2.  ] 

90.  aj2  +  4a;H-l  =  33.  99.    a;^  -  6aj- 3  =  13. 

91.  2/^ +  16 2/ +  54  =  90.  100.    a;- +  8a;- 2  =  18. 


COMMON   DIVISORS   OR   FACTORS. 


98.  1.  Name  a  common  divisor  or  factor  of  5  ic  and  10  xy. 
Of  4a6  and  16  a6. 

2.  Name  all  the  common  divisors  or  factors  of  24c  x^y^  and 
12ic^i/.  Which  of  these  is  the  highest  common  divisor  or 
factor  ?  Name  all  the  common  divisors  or  factors  of 
15  a^y^  and  25  ab^.  Which  is  the  highest  common  divisor  or 
factor  ? 

3.  What  prime  factors  or  divisors  are  common  to  24:X^y'^ 
and  12  :x?y  ?     To  15  a'h''  and  25  a6*  ? 

4.  How  may  the  highest  common  divisor  or  factor  be 
obtained  from  the  prime  factors  of  24  x^y'^  and  12  x^y  ?  How 
from  the  prime  factors  of  15  a^6^  and  25  aW  ? 

99.  An  exact  divisor  or  factor  of  two  or  more  quantities 
is  called  a  Common  Divisor  or  Factor  of  both  of  these  quan- 
tities. 

Thus,  3  a  is  a  common  divisor  or  factor  of  9  a?"  and  12  a. 

100.  The  divisor  or  factor  of  the  highest  degree  that  is  an 
exact  divisor  of  two  or  more  quantities  is  called  the  Highest 
Common  Divisor  or  Factor. 

Thus,  5  a^x  is  the  highest  common  divisor  of  20  a^x  and  15  a^x^. 

101.  Principle.  The  highest  common  divisor  or  factor  of 
two  or  more  quantities  is  equal  to  the  product  of  all  their 
common  prime  factors, 

85 


86  ELEMENTS  OF  ALGEBRA. 

102.  To  find  the  highest  common  divisor  or  factor  of  quan- 
tities that  can  be  factored  readily. 

1.  What  is  the  highest  common  divisor  of  4  a ^6  and 
12  a'b'c? 

PROCESS. 

4cr6    =2.  2.  a.  a.  6 
12a'b'c  =  3'2'2'a^a'b'b'C 
H.  C.  D.  =  2  .  2  .  a  .  a  .  6  =  4  a-6 

Explanation.  Since  the  highest  common  divisor  is  the  product 
of  all  the  common  prime  factors  (Prin.),  the  quantities  are  separated 
into  their  prime  factors.  The  only  prime  factors  common  to  the  given 
quantities  are  2,  2,  a,  a,  6  ;  and  their  product  is  4  a'^b.  Therefore  the 
highest  common  divisor  is  4  a^b. 

2.  What  is  the  highest  common  divisor  of  m{a^  —  b^) 
andm(a2-2a6  +  &')? 

Solution.  m{a?'  —  6^)  =  m(a  +  b)  (a  —  b) 

m{aP-  -2ab-\-  b'^)  =  m(a  -b){a-  b) 


H.  C.  D.  =  m  X  (a  -  5)  =  m  (a  -  6) 

Find  the  highest  common  divisor  or  factor  of  the  fol- 
lowing : 

3.  lA  QtFyz^  2ind  21  xyh. 

4.  18  xyz^  and  45  xyz, 

6.  5a^6-,  30  ab%  and  15  a^bc, 

6.  22mVz,Um%h\'dudl21mhi'z\ 

7.  20  abx^,  40  abV,  and  120  a^'bx^ 

8.  9  a^bmV,  27  b^m%  and  81  bmV. 

9.  a^~l  anda;2_2a:  +  l. 

10.  oc^  -\- 2  xy  -\-  y^  and  x^—  y\ 

11.  m^  4- ^^  and  m^  —  n^ 


COMMON   DIVISORS  OR  FACTORS.  87 

12.  a  —  h,  a?  —  h^,  and  a^  —  2ah-\-  h\ 

13.  x^ —  1  Siiid  x^  —  X'-2. 

14.  x^-2xdind2xi/-'4:i/, 

15.  yz  —  z  Sind  y^  —  1. 

16.  1  —  a^  and  1  +  a^. 

17.  3t^-\-2x  —  3^nda^-\-5x  +  6. 

18.  a?2_2a;-15anda;2  +  2a?-3. 

19.  a?2_3^._4an^^2_^._12. 

20.  x^  -l,x'-  1,  and  i»^  -  2  aj2  -f  1. 

21.  0^-707  +  6  and  a^  + 3  ic- 4. 

22.  aoj  +  bx,  a-m  —  ?>-m,  and  a^  -\-2ab-\-  61 

23.  a;'^  4-  2  ic  —  35  and  x^  -\-x  —  A2. 

24.  x^  —  4  ici/  +  ^2/^  and  05^  —  4  y"^. 

25.  aj2-8a?  +  15,  a.-2-4a;-5,  and  a;^  -  3  a;  -  10. 

26.  ^  +  x-20,x'-x-  12,  and  a;2  -  2  aj  -  8. 

27.  aj2  4-2a;i/  — 82/^  and  aj2  — 5a?2/4-6i/l 

28.  aj2  +  4a:2/  — 21i/2  and  aj^4-6a^2/ — '^2/^- 

29.  o^^^xy  —  1  y'^  mdx^  —  2xy-\-y'^. 

30.  a;2  — 4.x  — 5  and  a;2 +  2aj  — 35. 

31.  0.^  -  42/2,  ar^  +  4.xy  —  12y%  and  a:^  _  4.xy  +  4 2/^. 

32.  ax  —  3a,  a^^  _  j^j  + 12,  and  ax^  +  5aa;  —  24a. 

33.  am  +  2ma;,  a^  +  4aa;  +  4a^,  and  a^  —  2ax  —  Sx\ 

34.  b^  -  c^,  52  _|_  55c  4-  4c2,  and  6^  _  9&c  -  10c\ 

35.  a;2-2a;  +  l,  x2-8x  +  T,  anda;2-4a;+3. 

36.  a^  -  9,  a^  -  9a  -  36,  and  a^  -  7a  -  30. 

37.  4a  — 82/,  a2  —  5a2/  +  62/^  and  am —  2m2/. 

38.  2a; +  62/,  2{x^  +  6xy -{-9y^),  and  2 aa;  +  6 ay. 


COMMON   MULTIPLES.    "^ 

103.  1.  What  quantities  will  exactly  contain  3,  5,  x,  and  y  ? 

2.  What  quantities  will  exactly  contain  Sx-y^  and  9x^y? 

3.  By  what  quantity  must  9x^y  be  multiplied  so  that  it 
may  contain  3  x^y^  ? 

4.  By  what  quantities,  then,  must  the  highest  quantity 
be  multiplied  to  obtain  a  quantity  which  will  contain  each 
of  several  quantities  ? 

5.  By  what  quantities  must  the  highest  quantity  be 
multiplied  to  obtain  the  lowest  quantity  which  will  contain 
each  of  several  quantities  ? 

104.  A  quantity  that  will  exactly  contain  a  quantity  is 
cabled  a  Multiple  of  the  quantity. 

Thus,  a^x  is  a  multiple  of  a,  a'^,  a^,  x,  ax,  and  a^x. 

105.  The  lowest  quantity  that  will  exactly  contain  each 
of  two  or  more  quantities  is  called  the  Lowest  Common 
Multiple  of  the  quantities. 

Thus,  4  x'^y  is  the  lowest  common  multiple  of  4  x,  ?/,  and  x^. 

106.  Principle.  The  lowest  common  multiple  of  tivo  or 
more  quantities  is  equal  to  the  product  of  the  highest  quantity 
multiplied  by  all  the  factoids  of  the  other  quantities  not  con- 
tained in  the  highest  quantity. 

88 


COMMON  MULTIPLES.  89 

107.  1.  What  is  the  lowest  common  multiple  of  dx^yz* 
and  4:axyz^? 

PROCESS. 

5  oi^yz*  =  5  '  oc^  '  y  '  z* 
4  axyz^  =  4:  •  a  -  x  '  y  -  z^ 

L.  C.  M.  =  5  .  4  .  a  .  a^  .  2/  •  ^'  =  20aa^yz' 

Explanation.  Since  the  lowest  common  multiple  is  equal  to  the 
product  of  the  highest  quantity,  multiplied  by  the  factors  of  the  other 
quantities  not  found  in  the  highest  quantity  (Prin.)  for  convenience  in 
determining  what  factors  of  the  other  quantity  are  not  found  in  the 
higher,  the  quantities  are  separated  into  their  factors. 

Thus,  the  factors  of  the  lowest  common  multiple  are  seen  to  be  5,  4, 
a,  x^y  y,  z^j  and  their  product,  20«x"2?/^*,  is  their  lowest  common 
multiple. 

2.  What  is  the  lowest  corjimon  multiple  of  oj-  —  3  x  —  40 
and  0^24.  3a;  _  10? 

Solution.  x'^  -  3  ic  -  40  =  (x  -  8)  (a:  +  5) 

ic2  +  3x-  10=(x-2)(x  +  5) 


L.C.M.  zz:(x-8)(x+  6)(x-2) 
=  a:3-5x2-34x+80 


Find  the  lowest  common  multiple  of  the  following : 

3.  ^a?h''c  and  l^o?hc\ 

4.  lOaWx  and  15a%3(^. 

5.  Ua^bY^,  7bVy,  and  35abcx. 

6.  27  am,  33  a^my^  and  81  a V?/^. 

7.  15  oiPyh  and  21  x^y^z\ 

8.  iK-  —  4  and  a^  —  4 x  +  4. 

9.  x^  —  y^  and  x^  -{-2xy  -^  y^, 

10.  ar{x  —  y)  and  a{oi^  —  y^), 

11.  d'-b',  a2-h6^  and  a'-b^ 


90  ELEMENTS  OF   ALGEBRA. 

12.  a;2-9aj-22  and  aP --- 13x^22 . 

13.  a;2-h5aj  +  6  and  aj2^6a;+8. 

14.  x^  —  16,  a^  +  4a;4-4,  and  a.-^  — 4. 

15.  C-- 5c4- 4  and  c2-8c  +  16. 

16.  m(a  +  6),  m^(a  — &),  and  mx{a?  —  h'^), 

17.  a;(a'^  — ft*^),  a;(a  — 6),  and  x^ {a? -\- ah -\- y^) , 

18.  2a  +  l,  4cr— 1,  and  8a'^  +  L 

19.  a^  — a- 20  and  a^  4- a— 12. 

20.  x^y  —  xy'^y  X'  —  y'^,  and  x^  -f  ic^. 

21.  x^  —  x,  x^  —  1,  xy  —  y,  and  ab(x^  —  l). 

22.  ic'*  —  a^,  a^  —  a^,  and  x  —  a. 

23.  a^  -  5ab  +  W  and  a^  -  2  ab  +  61 

24.  X-  — i»-30  and  a;2+llaj  +  30. 

25.  a;2  — 1  and  .r^  +  1. 

26.  ic2  — 1  and  x2  +  4a;  -f- 3. 

27.  6(a4-6),  &2,  am(a  — 6),  and  a^ -{- 2 ab -\- b\ 

28.  15(a-6-a62),  21(a-^-a62),  and  35(a&2_^  6'^). 

29.  x^-i-5x-\-6,  x2-a;-12,  and  aj2-2a;-8. 

30.  0^-80^  +  15,  a.'2-4iK-5,  and  ^2-20^-3. 

31.  x^-\-4:xy  —  21y^,  x^  —  2xy  —  3y^,  and  x^  —  6xy  —  7 y^. 

32.  aj2-3a;-28,  ar^  +  x-12,  and  a^-lOx-^21, 

33.  a^  +  2a;-35,  oj^  +  a;  -  42,  and  a^  — 11  a; +  30. 

34.  a^  —  y\  x^ -]- xy -{- y^,  and  o^  —  y^. 

35.  a;2_3aj_4^  a.^-aj  — 12,  and  a;2  +  4a;  +  3. 

36.  a;^-l,  a;- -f- 1,  a;^  +  1,  and  x^-^l. 


FKACTIONS. 


108.  1.  When  an3^thing  is  divided  into  two  equal  parts, 
what  are  the  parts  called?  What,  when  it  is  divided  into 
three  equal  parts  ?  Into  four  equal  parts  ?  Into  m  equal 
parts  ?     Into  n  equal  parts  ? 

2.  What  does  -  represent  ?      -?      %?      -?      ~?      -? 

_„  ?     f? 

3.  Express  ^  of  a ;  ^  of  6 ;  ^  of  c ;  |  of  a ;  f  of  d. 

109.  One  or  more  of  the  equal  parts  of  anything  is  called 
a  Fraction. 

110.  A  quantity  no  part  of  which  is  in  the  form  of  a 
fraction  is  called  an  Entire  Quantity. 

Thus,  a,  3 X,  4  a  +  3  6,  etc.,  are  entire  quantities. 

111.  A  quantity  composed  of  an  entire  quantity  and  a 
fraction  is  called  a  Mixed  Quantity. 

Thus,  3  X  +  -^7   3  X  -  3  w  H-  ^  "^      are  mixed  quantities. 
5  c 

112.  The  sign  written  before  the  dividing  line  is  called 
the  Sign  of  the  Fraction. 

This  sign  belongs  to  the  fraction  as  a  whole  and  not  to 
either  the  numerator  or  the  denominator. 

Thus,  in  —  ^  "*"      the  sign  of  the  fraction  is  — ,  but  the  signs  of  the 

quantities  a,  ?>,  and  2  are  + .    The  sign  before  the  dividing  line  simply 
shows  whether  the  fraction  is  to  be  added  or  subtracted. 

91 


92  ELEMENTS   OF  ALGEBRA. 

REDUCTION  OP  FRACTIONS.  ] 

113.  To  reduce  fractions  to  higher  or  lower  terms.  \ 

1.  How  many  fourths  are  there  in  1  half?  In  3  halves?  ] 
In  5  halves  ?     In  a  halves  ?     In  b  halves  ?     In  n  halves  ?       \ 

2.  How  many  sixths  are  there  in  1  third  ?  In  2  thirds  ?  i 
In  8  thirds  ?    In  x  thirds  ?     In  y  thirds  ?    In  a  +  6  thirds  ?    i 

3.  Since  -  is  equal  to  — ;   and  -  is  equal  to  ^;  what    ■ 

may  be  done  to  the  terms  of  a  fraction  without  changing    ; 
the  value  of  the  fraction  ?  , 

4.  How  many  thirds  are  there  in  f  ?  How  many  halves  are  \ 
there  in  |  ?     How  tnany  fifths  are  there  in  f^  ?     How  many    ! 

twelfths  are  there  iu  —  ?    In  —  ?    In  —  ?  ■     '■ 

24  36  48 

5.  Change  — — ,    -— ,    -— ,    — -  to  fractions  whose  denom-    ! 

4       2      8  (X     16  j 

inator  is  16  a.  \ 

6.  Reduce  to  equivalent  fractions  whose  numerator  is  3  aj, 
J^     _3      ^     ^xy       xyz 

ax     by     al)     Axy^^    4:ayz  I 

7.  What  else,  besides  multiplying  them  by  the  same  1 
quantity,  may  be  done  to  the  terms  of  a  fractiou,  without  j 
changing  the  value  of  the  fraction  ?  j 

114.  A  fraction  is  expressed  in  its  Lowest  Terms  when 
its  numerator  and  denominator  have  no  common  divisor. 

115.  Principle.  Multiplying  or  dividing  both  terms  of  a 
fraction  by  the  same  quantity  does  riot  change  the  value  of  the 
fraction. 


FRACTIONS.  93 

116.    To  express  a  fraction  in  higher  terms. 

1 .    Change  ^ —  to  a  fraction  whose  denominator  is  15  6Vd. 
obc 

PROCESS.  Explanation.     Since  the  fraction  is 


2a 


to  be  changed  to  an  equivalent  fraction 
expressed  in  higher  terms,  both  terms  of 
56c  the  fraction  must  be  multiplied  by  the 

15b^c^d  -^  5bc  =  3  bed  same  quantity,  so  that  the  value  of  the 

2a  X  3  6cd       6 abed       ?^"^T  '''''^  T  ^^  ^^'^^S^^(}'^''^-)- 

y- — —  =         ^  In  order  to  produce  the  required  denom- 

OOC  X  6  oca      lob'Ca       inator,  the  given  denominator  must  be 
multiplied  by  3  bed ;  consequently  the 
numerator  must  also  be  multiplied  by  3  bed, 

2.  Change  — -  to  a  fraction  whose  denominator  is  30. 

5 

5  TYhYh 

3.  Change to  a  fraction  whose  denominator  is  24. 

4.  Change to  a  fraction  whose  denominator  is  28. 

5.  Change  ^  ^,^    to  a  fraction  whose   denominator   is 
45  6Vd2 


n  15b'c 


6.  Change  — — to    a    fraction  wliose  denominator 

is  33  a;.  ^^ 

7.  Change  — to  a  fraction  whose  denominator 

is  48.  ^^ 

8.  Change ^  to  a  fraction  whose  numerator  is  4a;-v. 

3a4-2 

9.  Change to  a   fraction   whose   numerator   is 

12abc.  ^''-'■^y 

10.    Change to  a  fraction  whose  denominator  is 

12xy.  ^ 


94  ELEMENTS  OF  ALGEBRA.  1 

7  .  .  i 

11.  Change to    a    fraction   whose    denominator   is  i 

w-  —  }r,  ' 

12.  Change  — -!— -  to   a   fraction   whose   denominator  is  ' 
9        o  m  —  71  \ 

nr  —  71"^.  • 

13.  Change ^^--  to  a  fraction  whose  denominator  is  75.  J 

1^  i 

1  .  *     ' 

14.  Change to  a  fraction  whose  numerator  is  m  +  n.  i 

m  -f  71  i 

i 

2a  .  i 

15.  Change to    a    fraction    whose    numerator    is  j 

2ax-4.a.       ^  +  2  t 

16.  Change     ~-^  to  a  fraction  whose  numerator  is  a;-  — - 1/^.  > 

j 
117.    To  express  a  fraction  in  its  lowest  terms.  ' 

15  x^v^  ' 

1.    Eeduce  -- — ^  to  its  lowest  terms.  ' 

20i»2/2  1 

PROCESS.  Explanation.     Since  the  fraction  is  to  be  changed  i 

.  p.    2  2       Q  to  an   equivalent  fraction   expressed  in  its  lowest  l 

y  =  zJ^  terms,  the  terms  of  the  fraction  may  be  divided  by  \ 
20  X'lf  4  any  quantity  that  will  exactly  divide  each  of  them  ^ 
(Prin.).  Dividing  by  the  quantity,  ^xy'^^  the  ex-  ] 
pression  is  reduced  to  its  lowest  terms,  since  the  terms  have  now  no  : 
common  factor.  Or,  the  terms  may  be  divided  by  their  highest  com-  ' 
mon  divisor.  \ 

Eeduce  the  following  to  their  lowest  terms :  ■ 

3a6c  13m^7i  17  m^nx^  ^ 

9a62*  '    39  mV'  *    Q^S  mnV*  • 

g       A.xyh  g     112a6^  ^     125aW  \ 

12  ^yz^'  '    252abxy'  '    625  aVz*'         \ 

^       10  abx  ^      35a^y'z\  ^^  2x  [ 

'    Sba'bcx^'  '    105  xYz''  '    4:X^-6ax       j 


FRACTIONS.  95 

Sab  -^      ax  — a  ar^  —  6  a;  —  40 

a-  —  ab^                        ax^  —a  '    oif  —  S x  —  70 

12.    ii±A.  18.       ^'  +  '^-^.  24.    "^-y". 

a'-b'                         a'-9a+U  a-'- -  f 

m  —  n                   -     ?ft-+8m  +  15  a^  —  1 

?)i-  —  rr                         711' —Zm  —  lo  .r"  + 1 

14.           «-^       ■  20.     '^•-2"'-^4.  26     J(^±lL. 

ft2-2a6+&-^                ar'-l2a;+36  27{a^-y') 

15           --g'-y-  21     ^'+^^-20  27       ^'^-^y 

x'+2xy+y'            '    scr  +  ix-S  '    8ar'-27/ 

16.      „  ^  +  ^      ■  22.    ^?^^.  28.       '^'-l 


aj^  +  2  o;  +1  m'^  —  7i^  2xy  -\-2y 

118.   To  reduce  an  entire  or  mixed  quantity  to  a  fraction. 

1 .  How  many  thirds  are  there  in  2  ?    In  5  ?    In  7  ?    In  a  ? 

2.  How  many  fifths  are  there  in  4  ?     In  7  ?     In  m  ? 

3.  How  many  fourths  are  there  in  3^  ?    In  5f  ?    In  a;  +  -  ? 

1 .   Eeduce  a;  -f  -  to  a  fractional  form. 


PROCESS.  Explanation.     Since   1  is  equal  to 

__  ^J(  ^,   aj    is   equal   to    ^ ;     consequently 


y  y 


^  I  a^xy  ^  a^xy^a       '^  y~  y  '^  y  y 

y     y     y        y 

EuLE.  Multiply  the  entire  part  by  the  denominator  of  the 
fraction  ;  to  this  product  odd  the  numerator  when  the  sign  of 
the  fraction  is  plus,  and  subtract  it  when  it  is  minus,  and 
write  the  result  over  the  denominator. 


96  ELEMENTS   OF  ALGEBRA. 

If  the  sign  of  the  fraction  is  - ,  the  signs  of  all  the  terms  in  the 
numerator  must  be  changed  when  it  is  subtracted. 

The  student  must  note  very  carefully  that  the  sign  of  the  fraction 
affects  the  whole  numerator  and  not  simply  the  first  term. 


Eeduce  the  following  to  fractional  forms  : 

2.    6x-{-^-^^  14.    X ^— 

2  7 


3.   4cf-^.  15.    3a;  + 

5 


6.    a.'  + 


7 


9 


.   8a-i^±^. 


,  a  —  h 
11.   x-\- 


Qf  a  —  X 


ax 
2  ac  -  c" 
a 

X 


4.    80.+^.  !«•    '*- 

4 

20,-1  18-   3+     ^ 


19.  3a + 


x'-l 
ab  —  a 


7.  2a;-^±i.  '' 

^  20.   a  +  a;  +  ^^±^. 

8.  7x  +  ^-^-^". 
6  ^^  ,       ,  2ac  — c^ 

21.    a  +  cH 

a  —  c 


5  22.    20.-3---^. 


a;  +  2 
3  23.   m  — 2rj  + 


10.   3mH 14 


m-\-2n 


c  24.    m  +  ^  — a;  + 

25.    a  — aj^- 


12     2x      y^^^  o.  ,  a2-|-aj2-5 

1^.       iiX •  OK  /7   O^  J ! . 


a-^-x 
13.    56  +  ^^-^^  26.    a(m  +  n)+    '^^^ 


m  —  n 


FRACTTOXS.  97 

119.    To  reduce  a  fraction  to  an  entire  or  mixed  quantity. 

1.  How  many  units  are  there  in  |  ?     In  |  ?     In  -2/  ? 

2.  How  many  units  are  there  in  ^ ?     In ? 

4       1  ^  ^ 

2 

1.    Reduce  ^^"^     to  a  mixed  quantity. 

X 

PROCESS.  Explanation.    Since   a  fraction   may  be  re- 

garded as  an  expression  of  unexecuted  division, 
^^  ~T  ^  =  a  -I-  -       ^y  performing  the  division  indicated,  the  fraction 
X  X       i^  changed  into  the  form  of  a  mixed  quantity. 

Reduce  the  following  to  entire  or  mixed  quantities : 

2.    23^.  10.    t±^. 

9  x-1 

26  a&  ^^     2a^-4&"^ 

'      11  *  '       a  4-  & 

4:5x^y^z  5  gy  +  ax  +  x 

15  xy  a 

^     36ac  +  4c  ^3     2ab  +  b\ 


9  a  +  6 

6.    ^'  +  ^  14.    ^^^'  +  ^ 
a  *     oj  — 4 

^^    12x^-5y  ^g^    ar^  +  2/^ 
6a;  x  —  y 

2a^x  —  ao[p  -^    2  a6  +  a6^  —  a' 


8.    ^^^^^^^^^ — ^=^-  16. 

a 


a6 


a—a?  a?^  —  x  —  1 

milne's  el.  of  alg. — 7. 


98  ELEMENTS  OF   ALGEBRA. 


18. 

2a2-2&2 

a  —  b 

19. 

x'-2x-{-l 

x^l 

20. 

2x'  +  5 
x-3 

21. 

a'-\-b' 

a  —  b 


22. 
23. 
24. 
25. 


X--2 

2x^-6x-{-4: 
2x  -3 

5x  —  1 
x^ -- a^  —  X -\- 1 


120.  To  reduce  dissimilar  to  similar  fractions. 

1.  Into  what  fractions  having  the  same  fractional  unit 
may  ^,  ^,  |  be  changed  ? 

2.  Into  what  fractions  having  the  same  fractional  unit 

may  —  and  be  changed  ? 

^  3m  2m  ^ 

3.  Express   - — ,  - — ,  and   - —    in   equivalent   fractions 

3m   2m  6m 

having  their  lowest  common  denominator. 

121.  Fractions  which  have  the  same  fractional  unit  are 
called  Similar  Fractions. 


122.  Fractions  which  have  not  the  same  fractional  unit 
are  called  Dissimilar  Fractions.  Similar  fractions  have, 
therefore,  a  common  denominator. 

123.  When  similar  fractions  are  expressed  in  their  lowest 
terms,  they  have  their  Lowest  Common  Denominator. 

124.  Principles.  1.  A  common  denominator  of  two  or 
more  fractions  is  a  common  multiple  of  their  denominators. 


FRACTIONS.  99 

2.    The  loivest  common  denominator  of  two  or  more  frac- 
tions is  the  loivest  common  multiple  of  their  denominators. 

1.    Keduce  —  and  — -  to  similar  fractions  having  their 

2x  (TX 

lowest  common  denominator. 

PROCESS.  Explanation.    Since  the  lowest  common 

o  9  denominator  of    several    fractions    is   the 

-     = — = lowest  common  multiple  of  their  denomi- 

LX      IX  X  a^      ^a-x      nators  (Prin.  2),  the  lowest  common  multi- 
3  3x2  6         pie  of  the  denominators  2  x  and  a'-x  must  be 

— ^  =  — ^ ^  =  o~2~      found,   which  is  2  a^x.     The  fractions  are 

ax  ax  X  ^  -  a-x  ^^^^  reduced  to  fractions  having  the  de- 
nominator 2  a^x,  by  multiplying  the  numer- 
ator and  denominator  of  each  fraction  by  the  quotient  of  2  a'^x 
divided  by  its  denominator.  2 «%  -^  2x  =  a^^  the  multiplier  of  the 
terms  of  the  first  fraction.  2 a^x  -=-  a'x  =  2,  the  multiplier  of  the 
terms  of  the  second  fraction. 


EuLE.  Find  the  lowest  common  midtiple  of  the  denomina- 
tors of  the  fractions  for  the  loivest  common  denominator. 

Divide  this  denominator  by  the  denominator  of  each  frac- 
tion^ and  multiply  the  terms  of  the  fraction  by  the  quotient. 

Eeduce  to  similar  fractions,  having  their  lowest  common 
denominator : 

2.  -  and  — 
be 

o     m       ;y  ab 

3.  —  and  — 
n  X 

,     ^xy       1    2  a 

4.  — ^  and 

c  '6  by 

5.  ^  and  ^. 

4  Zy 


6. 

2  b       .  3cd 
—  and 

xh           xz' 

7. 

ax     be     ab 

y     ax     ay 

8. 

be     ac     ab 

a,   — ?    — ?    — ' 

a^     ^z     yz 

9. 

2      2       2 

xy    ax    ayz 

100  ELEMENTS  OF   ALGEBRA. 

10  -^,  ^1.,  ^.  14  ^-±1.  ^-y  ^'  +  y' 

a?h     he     5V  •       4  2c  '       2a 


11.     ?      — ;r>      r—  15.      ?      > — ^ 

ojy       ar     xy-z  x  -{-  y     x  —  y     x'  —  y^ 

12.  H    H    %    i^.  16.    ^-.    ^- 


13      3      26     39  a  +  <^     a-5 

^^a-{-ba  —  bab  -^        3            5        9 

16.    1     >     — •  17.     ?     >     — • 

3aj         2?/        6  a;-f2a;-2a; 

a  +  &  1                 1 


18. 


19. 


20. 


21. 


aS  _.  &••     a  -  6  a-  +  a6  +  ^>' 

1  3  X 

x-i-l'    4  a; +  4  aj^  — 1* 

2  1  3 

a2_62'    ^_^'  a2  +  6=^* 

a;  +  2  a;-2 


of'  —  3  a;  +  2     ar'  H-  a?  —  2 

o»  n. 

22. 


23. 


24. 


25. 


aj2_^2a;-3     a)'^-2a?--15 

a;  +  l         a;  — 1        2  ^ 
a;(a;  — 2)     4a;  — 8     4aj 

5  3a;        4 -13a; 

14.2a;'    l-2a;'    l-4a.^* 

1  a  3  a 

a  +  b    a'  -  52'    a^  _  ft*' 


26.         ^-y       .     ^  +  y. 


a;2  —  a;?/  +  y^    ar^  4-  r    ^2/  +  ^^ 


FRACTIONS.  101 

CLEARING  EQUATIONS  OP  FRACTIONS. 

125.  1.  Five  is  one  half  of  what  number? 

2.  Eight  is  one  third  of  what  number?  One  fifth  of 
what  number? 

3.  li  ^x  equals  5,  what  is  the  value  of  a;? 

4.  If  ^  a;  =  6,  what  is  the  value  of  x  ? 

5.  What  effect  has  it  upon  the  equality  of  the  mem- 
bers of  an  equation  to  multiply  both  by  the  same  quantity  ? 

6.  If  the  members  of  the  equation  lx  =  6  are  multiplied 
by  5,  what  is  the  resulting  equation  ? 

7.  Multiply  the  following  equations  by  such  quantities 
as  will  change  them  into  equations  without  fractions : 

lx  =  S,  or  ^  =  8;   ix=7,  or  |  =  7; 

X  t  X  f\       X    ,    X        ^        X    ,    X         f-^ 

-  =  4:   —  =  ";   — —  =  b;    — —  =  o; 

8        '10         '2     4:        '36        ' 

?  +  *:5=10;  ^  +  ^  =  20;  -  +  -  =  8. 
4^8  '510  '3      5 

8.  How  may  an  equation  containing  fractions  be  changed 
into  an  equation  without  fractions? 

126.  Changing  an  equation  containing  fractions  into 
another  equation  without  fractions  is  called  Clearing  an 
Equation  of  Fractions. 

127.  Principle.  An  equation  may  be  cleared  of  fractions 
by  multiplying  each  member  by  some  multiple  of  the  denomina- 
tors of  the  fractions. 


102  ELEMENTS  OF  ALGEBRA. 


1.    Find  the  value  of  x  in  the  equation  a;  +  -  =  8- 

PROCESS. 

a^  +  ^  =  8 

•  Clearing  of  fractions,  3x-^  x  =  24: 
Uniting  terms,  4  a?  =  24 

Therefore,  x  =  6 

Explanation.  Since  the  equation  contains  a  fraction,  it  may  be 
cleared  of  fractions  by  multiplying  each  member  by  the  denominator 
of  the  fraction  (Prin.).  The  denominator  is  3  ;  therefore  each  mem- 
ber is  multiplied  by  3,  giving  as  a  resulting  equation  3  ic  +  ic  =  24. 
Uniting  similar  terms,  4  cc  =  24  ;  therefore,  x  =  6. 


2.    Find  the  value  of  x  in  the  equation 


,    X    ,    X    ,    X        tju 


PROCESS. 


4     2      5       2 
Clearing  of  fractions,  20x'\-5x-\-10x-{-4:X  =  390 
Therefore,  39  x  =  390 

and  a;  =  10 

Explanation.  Since  the  equation  may  be  cleared  of  fractions  by 
multiplying  by  some  multiple  of  the  denominators  (Prin.),  this  equa- 
tion may  be  cleared  of  fractions  by  multiplying  both  members  by  4, 
2,  5,  and  2  successively,  or  by  their  product,  or  by  any  multiple  of 
4,  2,  5,  and  2. 

Since  the  multiplier  will  be  the  smallest  v^hen  we  multiply  by  the 
lowest  common  multiple  of  the  denominators,  for  convenience  we 
multiply  both  members  by  20,  the  lowest  common  multiple  of  4,  2,  6, 
and  2.  Uniting  terms  and  dividing  by  the  coefficient  of  x,  the  result 
is  cc  =  10, 

Rule.  Multiply  both  members  of  the  equation  by  the  least. 
or  lowest,  common  multiple  of  the  denominators. 


FRACTIONS.  103 

1.  An  equation  may  also  be  cleared  of  fractions  by  multiplying 
each  member  by  the  product  of  all  the  denominators. 

2.  Multiplying  a  fraction  by  its  denominator  removes  the  de- 
nominator. 

3.  If  a  fraction  has  the  minus  sign  before  it,  the  signs  of  all  the 
terms  of  the  numerator  must  be  changed  when  the  denominator  is  re- 
moved. 

3.    Find  the  value  of  x  in  the  equation 
x  —  ^_x  —  l_ X  —  5 

Solution.  — ^^-^ 

Clearing  of  fractions,  9x  —  27  —  4ic4-4  =  6x  —  30 
Transposing,  9x  —  4aj— 6x  =  27  —  4  —  30 

Uniting  terms,  —x  =  —l 

Dividing  by  —  1,  x  =  l 

Note  3  under  the  rule  is  a  very  important  one.  Observe  its  appli- 
cation in  the  above  solution. 

Find  the  value  of  x  and  verify  the  result  in  the  following : 

4.  a;  +  ?  =  10.  10.   0.^4-^  +  1  =  11. 

4  Z      6 

5.  x  +  f  =  20.  .    11.   f  +  f  +  ^=9. 

3  3      4      6 

6.  2a;+f=9.  12.    2a;  +  ^+|  =  37. 

4  L    .       1 


X 


X  ,  X     69 


8.    4.  +  |=51.  14.    -  +  |  +  fo+5=^^- 

e.  |  +  3.  =  io.  15.  1  +  1+1  +  ^=17. 


104  ELEMENTS  OF   ALGEBRA. 

^9^12       12                        10       5  ^  2       4       4  ; 

17.  2a;  +  ?^4-||  =  27.             25.    ?+?L±^  =  x-3.  i 

4       15                             3        8  I 

18.  3a;-^  +  :^=70.                 26.    ^-'l^  =  x-d.  i 

o      iZ                                    3         11  * 

19.  xi----  =  3S.                     27.    — -2a^  =  ^^  +  ^-21  ^ 

7     5  7  5^ 

20.  0.  +  ^  +  ?^-?  =  ^         28.    ^Ii?+i  =  i5_2a..  '^ 

5        6       2      2                       3           3  ; 

21.  ll-^  +  ^  +  ^  =  s.            29.    2-^±^^2a;-^^±^.  i 

3       5^4^12                                      4                      3  i 

22.  2x  +  ^-^-^  +  ^  =  l^.        30.    gi+3^2^+3     7a;-5,  1 

8       3       6      2                      2            10     ^      o  i 

23.  a;  +  ^-^+^=59.            31.    7a;+16^x  +  8^^     '  ! 

2     4     11                                 21             21        3  i 

„„     a; +3    x  —  8_x  —  o     ^ 

4           5     ~     2          ■  i 

„„     6a;-14  ,  2a;-l      „^      .  i 

33.    — g i —  =  2a;-5. 

23a! +13     jg-l^g 

4          2"  ; 

„-     4a;  +  2      3a;-5  ,  .,  ] 

36.  ^±i  +  ^±3  =  ?i±i+16.  • 

3            4            5  , 

37.  ^  —  7  I  x  —  7  _x-{-l  'I 


10     '      5  6 


38     ^  +  3      x  —  l_x-\-42      a- 4- 5 


FRACTIONS.                                      10, 

39. 

2             3          6 

42. 

aj  ,  ic-2             5a:- 1 
5-^3="          6 

40. 

X     2.T-4      ^,      .r  +  T 
3          7      -"^         2    • 

43. 

a;      a:      2a;  _  a; -52 
8     5       5           4 

41. 

l-2.r     4-oa:      13 
3      ~      4          42* 

44. 

a;-.4      x-1      a;-26 
4             3             5 

45     ^-1 4.-^'-- 

X  — 

2     2 

^^-       2    +    4 

3 

3 

PROBIfEMS. 

128.  1.  A  certain  number  diminisliecl  by  1-  and  also  by  ^ 
of  itself  leaves  a  remainder  of  19.     What  is  the  number? 

Solution.     Let  x  =  the  number. 

Then  -  and  -  =  the  parts  of  the  number. 

5         6  ^ 

And  x----=19 

6      6 

Clearing  of  fractions,  30x  —  Ox  —  5x  =  570 

Uniting,  19x  =  570 

x  =  30 

2.  What  number  is  that  the  sum  of  whose  third  and 
fourth  parts  equals  its  sixth  part  plus  5  ? 

3.  A  man  left  i  of  his  property  to  his  wife,  i  to  his 
children,  and  the  remainder,  which  was  f  1200  to  a  public 
library.     What  was  the  value  of  his  property  ? 

4.  Out  of  a  cask  of  wine  I  part  leaked  away  ;  afterward 
10  gallons  were  drawn  out.  The  cask  was  then  |  full. 
How  many  gallons  did  it  hold  ? 

5.  A  man  leased  some  property  for  40  years.  ^  of  the 
time  the  lease  has  run  is  equal  to  ^  of  the  time  it  has  to 
run.     How  many  years  has  it  run  ? 

6.  The  sum  of  two  numbers  is  35,  and  ^  of  the  less  is 
equal  to  \  of  the  greater.     What  are  the  numbers  ? 


106  ELEMENTS  OF  ALGEBRA.  \ 

7.  A  man  paid  f  as  much  for  a  wagon  as  for  a  horse,  : 
and  the  price  of  the  horse  pUis  |-  of  the  price  of  the  wagon  \ 
was  100  dollars.     What  was  the  price  of  each  ?  i 

8.  The  sum  of  two  numbers  is  76,  and  |  of  the  less  is  | 
equal  to  \  of  the  greater.     What  are  the  numbers  ? 

9.  There  is  a  certain  number  the  sum  of  whose  fifth  ' 
and  seventh  parts  exceeds  the  difference  of  its  seventh  and  \ 
fourth  parts  by  99  ?     What  is  the  number  ?  ; 

10.  Divide  the  number  50  into  two  such  parts  that  \  of  ^ 
one  plus  f  of  the  other  shall  equal  35.  \ 

11.  A  son's  age  is  f  of  his  father's  ;  but  15  years  ago  the  '■ 
son's  age  was  \  of  the  father's.    What  are  the  ages  of  each  ?  \ 

12.  A  man  paid  f  3  a  head  for  some  sheep.  After  20  of  \ 
them  had  died,  he  sold  \  of  the  rjemainder  at  cost  for  f  60.  \ 
How  many  sheep  did  he  buy  ?  '    \ 

13.  A  man  wished  to  distribute  some  money  among  a  ; 
certain  number  of  children.  He  found  that,  if  he  gave  ; 
to  each  of  them  8  cents,  he  would  have  10  cents  left,  but,  if  a 
he  gave  to  each  10  cents,  he  lacked  10  cents  of  having  • 
enough.  How  many  children  were  there,  and  how  much  \ 
money  had  the  man  ?  \ 

14.  A  man  and  his  oldest  son  can  earn.  $30  a  week  ;  the  | 

man  and  his  youngest  son  can  earn  |^  as  much  ;  and  the  two  j 

sons  can  earn  $  19  a  week.     How  much  can  each  alone  earn  i 

per  week  ?  j 

! 

15.  A  and  B  start  in  business  with  the  same  capital.     A  j 

gains  $1775,  and  B  loses  $225.     B  then  has  |  as  much  j 
money  as  A.     What  was  their  original  capital  ? 

i 

16.  Divide  $440  between  A,  B,  and  C  so  that  B  shall  , 
have  $  5  more  than  A,  and  C  shall  have  f  as  much  as  A  ■ 
and  B.  i 


FRACTIONS.  lt)7 


ADDITION  AND  SUBTRACTION  OF  FRACTIONS. 
129.    1.    Find  the  value  of  i  +  i;  of  i  -  ^ ;  of  ^*  +  ^- 

2.  What  kind  of  fractions  can  be  added  or  subtracted 
without  changing  their  form  ? 

3.  What  must  be  done  to  dissimilar  fractions  before  they 
can  be  added  or  subtracted  ?  How  are  dissimilar  fractions 
made  similar  ? 


130.  Principles.  1.  Only  similar  fractions  can  he  added 
or  subtracted. 

2.  Dissimilar  fractions  must  be  reduced  to  similar  fractions 
before  they  can  be  united  by  addition  or  subtraction  into  one 
term, 

1.  What  is  the  sum  of  %  '^^  and  ^? 

2     5  4 

Solution. 
a     2a     3c_10a  .  8a     15c_18a4-15c 
2       6        4        20        20       20  20 

2.  Subtract  —  from  - — 

3x  Sy 

Solution. 
7_b_2a_  21  bx  _  16  ay  _  21  hx  -  16  ay 
Sy     Sx~  2ixy     2^xy  24  xy 

3.  Simplify  -  H —^ ^* 

^     ^  X     x-2     x-\-2 

Solution. 
2      x±\_x^_  2x'-S       a;3  +  3a;-H2x      x^~6x:'-  +  2x  ^  8a;-^-8 
X     x-2     x+2      x(x2-4)  x(x2-4)  x(x2-4)         x(x2-4) 


108  ELEMENTS  OF   ALGEBRA.  i 

i 

Eemember  to   change   the   signs   of  all  the   terms  in  the] 
numerator  when  you  subtract  a  fraction.  j 

Add:  \ 

4.    —  and  -^.  14.    -A_  and 


ah  ah  m  -f  n  m  —  n 

5.  2a6  ^„^  5«d.  i5_    OH- 1  ^^^^  a^_-^l. 
Zxy          3xy  x  —  1  x+1 

6.  ?and^^  16.    t|M^a.id-^. 
2/            71  x'—xy  x—y 

7.  ^andll.  17.    ^^^  and  ^-. 
3a          6  ax  xr  —y  x  —  y 

8.    and 18. and  — '—-' 

a-{-x  a  —  x  a-fJ  a  —  b 

9.  ^and^.  19.    ^i^and^JLh^. 

10.  2^  and  ^^.        20.    -^  and  l^Zll. 

3  c  12  c  2a;-l  4a;2_l 

2  »^  ,     2  a;  1  2 »; 

11.  ;^^  and  ^^.  21.    —5—  and  -=ii^. 

12.  ^and^.  22.    -      «         and         ^ 


a +  3  a +  5  a;(a  —  a;)  a(a  — .cj 

13.    l^^zll  and  2£zil/.  23.    AjL.  and      f'^^     ■ 
2,  X  —  y             x  —  y  a^x  a^  —  xr 

Subtract : 

24.  3-*  from  ^.  26.    '"-^  from  ^''. 

y  y  ax  y 

25.  I  from  ^.  27.    21  from  ^. 
o             o  xy  yz 


FRACTIONS.  109 


28.    -^— -  from -•  31.    — ^  from 


x-1             x-2  x-^2            x-\-3 

29.  ^^  from  -^.  32.    -^-^  from  A. 

X-  —  4:             x  —  2  it-  — 2  a;            2x 

30,  4^  from    .  ^  +  ^  ,.      33.    —^  from 


34.  Find  the  sum  of  one  half  of  c  dollars  and  one  third 
of  c  plus  d  dollars. 

35.  A  man  having  b  dollars  paid  out  one  half  of  his 
money,  and  then  one  fifth  of  the  remainder.  How  much 
did  he  pay  out  ?     How  much  had  he  left  ? 

36.  A  carpet  cost  a  dollars,  a  table  b  dollars,  and  a  desk 
one  half  as  much  as  the  carpet  and  the  table.  What  was 
the  cost  of  all  ? 

Simplify  : 

37     ?^  ,  4_xj/_3^  a-b     a-{-b     a      b 

*     15  "^    5  7  *  '       3     "^     4         2^5 

38.    ^x-l-x-\-\x.  X.      y      ,        x' 


39.   -f-^ab  —  \ab-{-\ab. 


45.    _  +  -^4--.^ 

y      x-\-y     x-4-xy 


.  xt/ xy  1 

40.  ^  +  ^_^.  ^'  x^y      x-y'^x^y 

1       ,       1       ,      2a 

..     3a;  ,  a;  ,  3a;  47.  -7——  +  Z +  :j 

41.  —-!--+— -.  l-f-a      1-a      1-a- 

5       4      10 

2a__46      a  48.  a-b  ^b-c      c-a 

'     7        9      3  ab  be  ac 

.       5  ax     2  ax     5  ax  .^      2x 3  5 

'      6        ~9"       12  *  a;2-l      a;  +  l      a;-l* 


110  ELEMENTS  OF  ALGEBRA. 


50.    4^^.--^.  +  -^-        53.    .-^  +  - 


a;2— x— 2     x—2     x-\-l  a;  4-1      x  —  1 

51.    -H -H 64. 1 

x-\-l      X  -\-2      X  —  3  x  —  a     x-^-a      x 

^^  1  1  ^^        a      ,     3a  2  ax 

5/3.    —^ -— -•  55. h 


x^  —  X  —  2     x^ -{-X  —  2  a  —  X     a  -{-  x     a^  -{-x^ 

56.  A  man  walked  a  miles  the  first  day ;  the  second  day 
he  walked  half  as  far  as  the  first  day ;  and  the  third  day 
half  as  far  as  the  second.  How  far  did  he  walk  in  the 
3  days  ? 

57.  What  is  the  cost  of  a  coat,  hat,  and  pair  of  shoes,  if 
the  coat  costs  a  dollars,  the  hat  c  dollars,  and  the  shoes  -|  as 
much  as  the  hat  and  coat  ? 

58.  A  train  went  e  miles  the  first  hour, /miles  the  second, 
and  then  went  back  ^  of  the  whole  distance.  How  far  was 
the  train  then  from  the  starting  point  ? 

MULTIPLICATION  OF  FRACTIONS. 
131.     1.    How  much  is  three  times  ^?     Four  times  f  ? 

Five  times  —  ? 
9 

2.  Express  2  x  f  in  its  lowest  terms;  5  x  y^-;  8  x  g^j 
3x— . 

3.  How  may  the  products  in  1  and  2  be  obtained  from 
the  given  fractions  ? 

4.  In  what  two  ways,  then,  may  a  fraction  be  multiplied 
by  an  integer  ? 

5.  How  much  is  1  of  4  or  4 -3?     i  of  i^  or  i? -- 2? 


FRACTIONS.  Ill 

6.  In  what  two  ways,  then,  may  a  fraction  be  divided  by 
an  integer  ? 

132.  Principles.  1.  Multiplying  the  numerator  or  divid- 
ing the  denominator  of  a  fraction  by  any  quantity,  multiplies 
the  fraction  by  that  quantity. 

2.  Dividing  the  numerator  or  multiplying  the  denominator 
of  a  fraction  by  any  quantity,  divides  the  fraction  by  that 
quantity, 

1.   What  is  the  product  of  -  multiplied  by  -? 

PROCESS.  Explanation.     To  multiply   -   by  ^  is  to  find  c 

X      c       ex       times  J  part  of  -•     J  part  of  -  is  —  (Prin.  2),  and  c 

^.  ^      times  -^  is   —  (Prin.  1).    That  is,  the  product  of  the 

Sy        Hy 

numerators  is  the  numerator  of  the  product,  and  the  product  of  the 

denominators  is  the  denominator  of  the  product. 

1.  Reduce  all  entire  and  mixed  quantities  to  the  fractional  form 
before  multiplying. 

2.  Entire  quantities  may  be  expressed  in  the  form  of  a  fraction  by 
writing  1  as  a  denominator.     Thus,  a  may  be  written  -• 

3.  When  possible,  factor  the  terms  of  the  fractions  and  cancel 
equal  factors  from  numerator  and  denominator. 

2.    Find  the  product  of  — — -  x  — X  ^ 


a^_4        4a;        x^^2x-3 


Solution. x  — ^^ x 


x2-4        4x        x'^-2x-3 

x-^  (x+l)(x-l)      x±2_ 


X  ^ ■ f-^ '-    X 


(x-f2)(x-2)  4x  (x-3)(x+l) 

Canceling  equal  factors  from  dividend  and  divisor,  the  result  is 
x-\ 
4xCx-2)' 


112  ELEMENTS  OF   ALGEBRA. 

Multiply : 

3.    by  abn.  9.    by 

abcx  18  m  +  n 

4.  ;^;^ — "2  by  6  a.  10.    ^^ ^bya;+2/. 

5.  n-by--  11. 

6.  - — by 12. 

0  ax       xyz 

„     7  am  1     3  mn 

7.  — -—  by 13. 

9  bn         2  ax 

8.  ^±lby^.  14.  _,     , 

a6c         aa;  5a;—  15       ar  +  5x4-4 

a^  —  ab       ,     a^  —  4  aft  +  3  &^ 


a^-2ab  +  b''    '  a' -J)' 

,-       o;^  — ic  — 2i     ic^  — 3a;  — 10 
16.    by  . 

aj2^4a.-  +  3    ^        a^-4 
3m -12     ,  m4-2 

17-  ~TT-^ — T-^^y 


?7i^  +  3  m  +  2       m'^  —  5  m  4-  4 
18.    /  +  ^    by      ^-^ 


5aj-15    *'a^-4aj-21 

20      ,     ar^  +  3a;-40 

19.    by  — ' • 

aj2_64    J'      5  a; -25 

20.  ^l#by     "^' 


2^     a;^  +  2a;-63v     a;^  +  9a;4-18 
a;2_4^_21    ^      6a;  +  54 


FRACTIONS.  113 


Simplify  the  following : 

22.  ^x4x^. 
DC     ah      ac 

23.  ^-=l!x-^X^^. 

X  x+y      x—y 

24.  (2--y  4 


25.       -+- 


1  ,  ly   x^y 


X     yj  \xf  —  y 


26.    (—^A^\(t^Z^\(tj^\ 
\^-\-\^x-^1h)\  x-2  J\x-5j 

'x  ,  X     2y\f      ^xy 


5  aj-  —  3  xy^ 

28.    i^  X    ^^^    X  ^^^'^y^  X  ^"-^ 
9  0^2;       ^  —  y^         8  ic?/^  a?2/2; 

\mr  -\-m  -\-  \j\mr      m 

30.^X^X^^^20-15. 
a  +  3      a— o        a-  —  a— 2 

31.  If  a  cords  of  wood  cost  h  dollars,  how  much  will  c 
cords  cost  at  the  same  rate  ? 

32.  If  &  pounds  of  sugar  are  worth  c  pounds  of  butter, 
how  many  pounds  of  butter  are  6  pounds  of  sugar  worth  ? 

33.  A  father  works  a  weeks  at  n  dollars  per  week,  and 
his  son  works  4  weeks  at  ^  of  the  wages  that  the  father 
receives.     How  much  do  both  receive  ? 

34.  What  is  the  value  of  d  acres  of  land  at  f  of  {d  —  e) 
dollars  per  acre  ? 

35.  A  walked  c  miles  in  b  hours.  How  far  did  he  walk 
in  1  hour  ?     In  5  hours  ?     In  a  hours  ?     In  c  +  d  hours  ? 

milne's  el.  of  alg.  —  8. 


114  ELEMENTS  OF  ALGEBRA. 


DIVISION  OP  FRACTIONS. 
133.    1.   How  many  times  is  ^  contained  in  1  ?    i  in  1  ? 
1    ml?     yViiil^     -inl?     ;i-  in  1  ?      -J_  in  1  ? 

2.  How  does  the  number  of  times  a  fraction,  having  1 
for  its  numerator,  is  contained  in  1  compare  with  its 
denominator  ? 

3.  How  many  times   is   f  contained   in   1  ?     |   in   1  ? 

^  in  1  ?     ?  in  1  ? 
c  a 

1.  What  is  the  value  of  —  -^  -  ? 

2c     d 

PROCESS  Explanation.     The  fraction  -  is  con- 

a  ^^ 

b    ^a       b       d       bd      tained  in  1  d  times;  and  ^  is  contained  in. 

2c     d     2  c     a     2ac    ^l  p^j.^  ^f  ^  ^^^^^^  ^^  d  ^^^^^     ^^^^ 

-.a  ^  h        h 

since  -  is  contained  in  1   -  times,  it  will  be  contained  in  — ,    — 
d  a  2c     2c 

times  -  or  -^  times.     That  is,  the  quotient  of  one  fraction  divided 

a        2ac 
by  another  is  equal  to  the  product  of  the  dividend  by  the  divisor 
inverted. 

,1.    Change  entire  and  mixed  quantities  to  the  fractional  form. 

2.  An  entire  quantity  may  be  expressed  as  a  fraction  by  writing  1 
for  its  denominator. 

3.  When  possible,  factor  the  terms  of  the  fractions  and  use 
cancellation. 


2.   Divide    -^-Tli—by   '"^^ 


x^-\-x-20    '  x'-25 

Solution.         x^  -  1       ^  .x+j_  ^  (x  +  l)(x  -  1)  ^  (x+5)(x-5) 

x2  +  a;-20     5c2-25      (x  +  5)(ac-4)  a:  +  1 

Canceling  equal  factors,  the  quotient  is 

(a;-l)(y. -5)  ^^  x^-6x  +  b 

X  —  4  X  —  4: 


FRACTIONS.  115 


Divide : 

3-  -  by  -•  12.  -Hb  by  — ^ 


2/       n  ^  —  y^      ^-\-y 

^  by  ^.  13.    *i 

xy         c  2y 


10. 


11. 


^'   6a'b'c 


9. 


13  a; +  42 


•    a;^_2a:-15    '^  x'-{-Ax-^3 
23^    a^-lOa  +  24^     /a-6^       a-.4 


a^+oa- 14     -    \a-2      a^^6a-7 

V2^3;     "^   V      6(0^  +  2/) 
25.    4:x'-4.y'  by  '^^-i^- 


116  ELEMENTS  OF  ALGEBRA. 

26.    4=4  by  ^SZV.. 

28.  What  is  the  cost  of  h  tons  of  coal  if  7  tons  cost  a 
dollars  ? 

29.  How  many  times  a  dollars  are  2  times  c  dollars  ? 

30.  A  man  sells  4  horses  at  m  dollars  each,  and  after 
using  n  dollars  of  the  money,  divides  the  remainder  equally 
among  c  children.    What  represents  the  share  of  each  child  ? 

31.  A  man  exchanges  d  horses  for/  cows.  At  that  rate 
what  is  the  value  of  a  horses  ? 

32.  A  son's  age  is  —  of  his  father's.     The  father's  age 

m 

is   represented  by  f  of  the  difference  between  a  and  h. 
What  represents  the  age  of  the  son? 


EQUATIONS. 
134.     Elimination  by  Comparison. 

1.  cc  -f  3  2/  =  5  and  x  —  y=l  are  simultaneous  equations. 
Transpose  the  term  containing  y  in  each  to  the  second 
member.     What  are  the  resulting  equations  ? 

2.  Since  the  second  members  of  these  derived  equations 
are  each  equal  to  x,  how  do  they  compare  with  each  other? 
(Ax.  1.) 

3.  If  these  second  members  are  formed  into  an  equation, 
how  many  unknown  quantities  does  it  contain  ? 

4.  How,  then,  may  an  unknown  quantity  be  eliminated 
from  two  simultaneous  equations  by  comparison  ? 


EQUATIONS.  117 

1.    Given  ^  !"        -' '    ^^  >•  to  find  the  values  of  a;  and  v 
by  comparison. 


2x-y=S 

(1) 

x  +  Sy=zl9 

(2) 

2x  =  'S  +  y 

.(3) 

.=i±i 

(4) 

x=lQ-Zy' 

(5) 

3  +  2'=19      32, 
2                    " 

(6) 

3  +  2/  =  38 -62/ 

(7) 

72/ =  35 

(8) 

2/  =  5 

(9) 

a;  =  19  -  15 

(10) 

a;  =  4 

(11) 

Solution. 

Transposing  in  (1), 

Dividing  by  the  coefficient  of  x, 

Transposing  in  (2), 

Equating  (4)  and  (5)  (Ax.  1), 

Clearing  of  fractions, 
Transposing,  etc., 
Dividing  by  the  coefficient  of  ?/, 
Substituting  the  value  of  y  in  (6), 


EuLE.  Fmd  an  expression  for  the  value  of  the  same 
unknown  quantity  in  each  equation.  Make  an  equation  of 
these  values  and  solve  it. 

Solve  the  following  by  elimination  by  comparison  : 
^      |4aj  +  2/  =  10|  ^      (2x-2y  =  -Q^ 

\    x  +  y=    7)  '     \3x-2y=      6)      • 

•     (    a;  +  3.y  =  32i  *     \2x-y  =  22) 

(X    -\-2y  =  ll>^  (3x-    y  =  U^ 

4.     ■<  >•  8.     s  >■      - 

(6a;-    2/  =  40)  \2x-^2y  =  2%\ 

^      (    x  +  lOy^l^^  ^      (3a:+    y  =  26| 

(2a;-      2/=    7>  *     1    x-\-2y  =  21) 


118 


ELEMENTS  OF   ALGEBRA. 


10. 


11. 


12. 


13. 


14. 


16. 


16.     { 


\2x-\-3y  =  2o\ 
(2x-    y=    7  I 
(3a;-f  2i/  =  28  3 
(Zx  —  2y  —  2^ 
\2x-    y  =  b) 
(4aj-3?/=    1| 
\2x+    y  =  23\ 
(2x^y  =  27^ 
(3aj-2/=    3) 
(3a;-f  2i/  =  30| 
\bx-{-    2/ =  29) 

r  6  8 

I     ' 


X     y 
X       y 


17. 


18. 


19. 


20. 


{ 


ax-\-hy  =^m^ 
=  n  ) 


ax-\-cy  •• 


2      3" 


5 


{^2x-\-  ay  =  b 
\  ax  -\-2y  =  c  I 


3     2 
9      2 


21.     <^ 


o 

?  +  5a;  =  27 


135.    Elimination  by  Substitution. 

1.  In  the  equation  x-\-  y  =  12,  how  may  the  value  of  x 
be  found  if  y  equals  5  ? 

2.  a;  4- 22/ =  8  and  2  a?  4-3?/ =  13  are  simultaneous  equa- 
tions. Find  the  value  of  x  in  the  first  equation  by  trans- 
posing y  to  the  second  member. 

3.  If  this  value  of  x  should  be  substituted  for  x  in  the 
second  equation,  how  many  unknown  quantities  would  the 
resulting  equation  contain  ? 

4.  How,  then,  may  an  unknown  quantity  be  eliminated 
from  two  simultaneous  equations  by  substitution  ? 


EQUATIONS.  119 

1.    Given  <        ~  ,    "  .     ^  to  lind  the  values  of  x  and  v 

by  substitution. 

Solution.                                    Sx  —  2y  =:  1  (1) 

X  +  4  y  =  19  (2) 

From  (1),                                               x  =  ^  "^^^  (3) 

o 

Substituting  (3)  in  (2),     J-±-?l  +  4  ?/  =  19  (4) 
o 

Clearing  of  fractions,       1  +  2  ?/  +  12  ?/  =  57  (5) 

Transposing,  etc.,                             14?/ =  56  (6) 

2/ =  4  (7) 

Substituting  in  (3),                               x  =  ^—^  (8) 

o 

X  =  3  (9) 


Solve  the  following  by  elimination  by  substitution : 

^      (     x-3y=    3j  ^      r7a;-4^=81| 

(20^  +  4?/  =  16  i  *     (2x-     2/  =  25i 

r3a;-2iy=3|  i     x-\-2y  =  20^ 

(207  +  32/  =  28 1  ^*     i     x-'2y=    4:1 

5      (4^  +  42/=76i  ^^      r3a.-    y  =  5, 

\Sx-{-    2/=39>  *     (32/-    x  =  9> 

(3x-{-    2/  =  14|  ^^      f2aj-     y=      18  | 

(    aj  +  32/  =  26)  '     I     a;-22/=-6J 


120 


12. 


13. 


14. 


15. 


16. 


ELEMENTS  OF   ALGEBRA. 


\2x-     y=20} 


I  6  a;  +  5  ?/  =  180  3 

^  5  a;  +  3  ?/  =  46 1 

\Sx  +  2y  =  29i 

(    2x-     y=    10 1 
(16a;-52/  =  20ol 


x  +  l=ie] 


[  +  y  =  20 


17. 


18. 


19. 


20. 


(4:X-I5y  =  24:^ 
[3x-2y  =  19\ 


3x-2y 

-+7  =  3 
a     b 


X     y 
a     b 


{  2jo 
3  ' 


2 

y. 

7 


:  =  o 


X  ,  y 


a 

.  b 

+  -  = 

m 

X 

y 

> 

G 

.  d 

+  -  = 

n 

X 

y 

REVIEW  OP  EQUATIONS. 

136.  1.    Definition  of  an  Equation. 

2.  Definition  of  Members  of  an  Equation. 

3.  Definition  of  Transposition.     Eule. 

4.  Definition  of  an  Axiom.     Give  the  Axioms. 

5.  Definition  of  Simultaneous  Equations. 

6.  Definition  of  Elimination. 

7.  Definition  of  Clearing  of  Fractions.     Rnle. 

137.  The  highest  number  of  factors  of  unknown  quanti- 
ties contained  in  any  term  determines  the  Degree  of  an 
Equation. 

Thus,  when  only  one  unknown  quantity  of  the  first  power  is  found 
in  any  term,  the  equation  is  said  to  be  of  the  first  degree ;  when  the 
second  power  of  an  unknown  quantity  is  found  in  any  term  or  when 
the  number  of  unknown  factors  in  any  term  is  two,  the  equation  is 
said  to  be  of  the  second  degree,  etc. 


EQUATIONS.  121 

138.  An   equation  of  the  first  degree  is  also   called  a 
Simple  Equation. 

Thus,  x-\-y=li^ax-\-2  a^y  =  10  are  simple  equations. 

139.  An  equation  of  the  second  degree  is  also  called  a 
Quadratic  Equation. 

Thus,  x2  =  c,  X  -\-  xy  =  4:,  abcxy  =  6  are  quadratic  equations. 

140.  Solve  the  following  equations  : 

1.  x-6-{-5x=4:-{-2x-{-2. 

2.  4:{x-l)=:5{2x-S). 

3.  x{x-4:)-{-6  =  (x-^)(x  +  30). 

4.  (2x-^6)(x-\-l)  =  {2x-2){x-{'9), 

5.  ax  —  bx  =  cr  —  b^. 

6.  ax  —  ab  =  x  —  ex  -f  b'\ 

7.  a  ■{■  b -{- 2  X  =  c^x. 

8.  a^x  —  a  —  b  =  b^x  -{-  2  ab. 

13.  ^-x  =  ^-^-9. 
3  11 

14.  ^(20.^-6)  =|(a^4-4)-7. 

15     11  ^ -  80  _  Sx-5  ^  ^ 
6  15     ~    * 

16.    ^^3^3(a:  +  3)^3(..  +  3)_l, 

7  2  2 


122  ELEMENTS  OF   ALGEBRA. 


17. 

5             9         2 

18. 

X       ,       c 
a             d 

19. 

ax  —  h,^      x-{-ac 

\      Uj  

C                                C 

20. 

1             1            2a; 
x-1      x^l      x""-! 

21. 

4       ,     12    _    48 
x-2      x-\-2     i»2_4 

22. 

a;_12ic-4_^         7 
0^-7       x-- 12      '^      x-1 

23. 

X     x^  —  6x      2 
3      ^x-1      3 

24. 

aj  +  7      i»  +  o_^      5a; +  3 
2             3     ~              4 

25. 

^^^  +  ^^-^^'-^  =  10(x-l). 

oa 

7  a; +  16      a;        x-\-^ 

21  3      4a; -11 

Suggestion.     The  equation  may  be  expressed  as  follows : 
7x16      x_x  +  8 


Simplifying, 


21      21      3     4x-ll 
16^    x  +  S 
21      4x-ll 


20a;     36      5a; +  20^  4a;     86 
25       25      9a;-16       5       25* 


28.  1 ^=_:^. 

a;-|-2      x-\-Q 


EQUATIONS.  '  128 


29. 


30. 


31. 


5a;     ,     10        ^ 

x  +  6^  x-1-^- 

15               1            44 

x  +  3     3(a;  +  3)      15 

3a; -2      6a; +13  _  21- 

-3x 

2x-5  10 


32.       7         6a;-1^3(16-2a^) 
X  —  1        X  -\-l  a^  —  1 

32     10  0^  +  17  _  12a;  4- 2  ^  5a?-4 
18         ,  13X--16  9 

See  suggestion,  Example  26. 
_.     18^  +  19  ,  lliu-f  21_9a.'  +  15 


35. 


28  6ir  +  14  14 

6a?  +  l       2a;-4  ^2a;-l 
15  7a.'-16~       5 


36     ^  ^'  —  -^  ,  ^'  4-  5  ^  g;  4- 13       1 
16  4  8  16' 

4  0^  +  3  ^  8  37  +  19  _  7  a? -29 
9      ~       18  5a;- 12* 

38.    -±-^      '  ^' 


x-^2      x-^  '3      a;-  +  5a;  +  6 
39.    2^1^  +  1      1-^^ 


40. 


l_3a;      6      l-2a; 
4  10 


a;  -  4      x-5      a;^  -  9 a;  +  20 


b  a 

42.   -^-t-A  =  a2  +  6^ 
bx     ax 


124 


46. 


47. 


43. 


44. 


ELEMENTS  OF   ALGEBRA. 

x'^—  a  _  a^x  _2x  _a 
bx  b  h       X 

Sax --2  b      ax  — a     ax     2 


45 


3b  2b 

b_(l±^^,^^b_x^ 


ex 


(     x-^5y=    7 
X4tx-^3y 


-'I 
=11) 


\3x  +  3y='d6\ 


x-y=8 

9 


48.     i 


49.     < 


^-•^  =  3 


^  +  •^-=8 
5     2 


52. 


x-\-hy=Vl' 
l+3y=    7 


53. 


Xx  —  y=  b) 


54. 


55. 


^  ax-{-by=  c  ^ 
\bx  —  ay  =  d) 


a 

,  h 

- 

+  -  = 

m 

X 

y 

a 

b 

V 

— 

:=: 

?i 

X 

y 

50.     i 


3       8 
X     y  _       ^ 


56. 


— \-  7iy  =  m  -{-  n 
m 

mx  ,  ii^v         9  ,     2 
n        m 


51. 


^4-^  =  2 
2      2 

7  +  ^  =  2^ 


57. 


a     b  I 

1^6     a         J 


EQUATIONS.  125 

PROBLEMS. 

141.  1.  The  sum  of  two  numbers  is  100,  and  the  less  num- 
ber is  10  more  than  i  of  the  greater.    What  are  the  numbers  ? 

2.  A,  B,  and  C  build  216  rods  of  fence.  A  builds  7  rods 
a  day,  B  builds  6  rods,  and  C  builds  5^  rods.  If  A  and  B 
work  the  same  number  of  days  and  C  works  twice  as  many 
days  as  the  others,  how  many  days  does  each  work  ? 

3.  A  man  has  $  6000  in  cash.  He  expends  part  of  the 
sum  for  a  house,  and  invests  the  remainder  where  it  brings 
him  an  income  of  f  250  per  year.  If  in  4  years  the  amount 
invested  together  with  the  income  of  that  time  equals  the 
amount  paid  for  the  house,  how  much  is  paid  for  the 
house  ?     How  much  is  invested  ? 

4.  A  boy  bought  a  certain  number  of  apples  at  the  rate 
of  3  for  2  cents.  He  sold  them  at  the  rate  of  6  for  5  cents, 
and  gained  12  cents.     How  many  apples  did  he  buy  ? 

5.  A  lady  bought  two  pieces  of  cloth;  the  longer  piece 
lacked  4  yards  of  being  three  times  the  length  of  the 
shorter.  She  paid  f  2  per  yard  for  the  longer  piece,  and 
f  2^  for  the  shorter,  and  the  shorter  piece  cost  just  ^  as 
much  as  the  longer.  How  many  yards  were  there  in  each' 
piece  ? 

6.  The  distance  around  a  rectangular  field  is  equal  to 
10  rods  more  than  five  times  the  breadth  ;  and  the  length  is 
1|  times  the  breadth.  What  are  the  length  and  the  breadth 
of  the  field  ? 

7.  A  man  spends  J  of  his  annual  income  for  his  board, 
i-  for  traveling  expenses,  -^^  for  clothes,  ^  for  other  ex- 
penses, and  saves  f  380.     What  is  his  annual  income  ? 

8.  Of  a  company  of  soldiers  f  are  on  duty,  ^  of  the 
remainder  are  absent  on  leave,  -^^  of  the  whole  are  sick,  and 


126  ELEMENTS  OF  ALGEBRA. 

the  remaining  50  are  off   duty.     How  many  soldiers  are 
there  in  the  company  ? 

9.  A  farm  of  450  acres  was  divided  between  A^  B,  and 
C  so  that  A  had  f  as  many  acres  as  B,  and  C  had  i  as  many 
acres  as  A  and  B  together.     How  many  acres  had  each  ? 

10.  A  boy  bought  a  certain  number  of  apples  at  the  rate 
of  2  for  1  cent.  He  sold  half  of  the  number  for  1  cent 
apiece,  and  the  other  half  at  the  rate  of  2  for  1  cent.  He 
gained  10  cents  by  the  transaction.  How  many  apples  did 
he  buy  ?  '  ' 

Suggestion.    Let  2x  equal  the  number  of  apples. 

11.  The  length  of  a  certain  field  is  twice  the  breadth. 
If  the  length  were  |  as  much,  and  the  breadth  li-  as  much, 
the  entire  distance  around  the  field  would  be  64  rods.  What 
are  the  length  and  the  breadth  of  the  field  ? 

12.  Find  a  number  whose  half,  third,  and  fourth  added 
together  equal  the  number  plus  2. 

13.  A  man  in  business  gained  $  100,  and  then  lost  ^  of 
all  that  he  had.  He  afterward  gained  $  150,  when  he  found 
that  his  money  was  just  equal  to  his  original  capital.  What 
was  the  original  capital  ? 

14.  Two  numbers  are  to  each  other  as  3  to  4.  If  10  is 
subtracted  from  each,  the  smaller  one  will  be  |  of  the  larger. 
What  are  the  numbers  ? 

Suggestion.    Let  3  x  and  4  x  represent  the  numbers. 

15.  A  woman  sold  some  eggs  at  2  cents  apiece  ;  but  when 
she  came  to  deliver  them,  four  of  the  eggs  being  broken, 
she  received  only  192  cents.     How  many  eggs  had  she  ? 

16.  Two  numbers  are  to  each  other  as  5  to  7.  One  half 
of  the  first  plus  the  second  equals  one  half  of  the  second 
plus  12.     What  are  the  numbers  ? 


EQUATIONS.  127 

17.  A  said  to  B,  "Give  me  $100,  and  I  shall  have  as 
much  money  as  you  have  left."  B  said  to  A,  "Give  me 
$  100,  and  I  shall  have  three  times  as  much  money  as  you 
have  left."     How  much  money  had  each  ? 

18.  A  man  gained  in  business  as  follows :  The  first  year 
$  400  less  than  his  original  capital ;  the  second  year  $  300 
less  than  the  original  capital ;  and  the  third  year  $  200  less 
than  the  first  year.  The  gains  of  the  three  years  amounted 
to  f  700  more  than  the  capital.  What  was  the  original 
capital  ? 

19.  A  mason,  5  carpenters,  and  3  assistants  receive 
together  f  191  for  working  a  certain  number  of  days.  The 
mason  receives  f  3  per  day,  the  carpenters  f  2^,  and  the 
assistants  $  1^.     How  many  days  do  they  work  ? 

20.  A  man  spent  $  300  more  than  i  of  his  earnings  each 
year.  In  5  years  he  had  saved  f  1000.  How  much  did  he 
earn  each  year  ? 

21.  A  boy  bought  a  certain  number  of  apples  at  the  rate 
of  4  for  5  cents,  and  sold  them  at  the  rate  of  3  for  4  cents. 
He  gained  60  cents.     How  many  did  he  buy  ? 

22.  A  boy  spent  ^  of  his  money  and  2  cents  more.  He 
then  spent  i  of  what  was  left  and  2  cents  more,  when  he 
found  that  there  remained  12  cents  of  his  money.  How 
much  had  he  at  first  ? 

23.  A  man  bought  a  horse  and  wagon,  paying  twice  as 
much  for  the  horse  as  for  the  wagon.  He  sold  the  horse  at 
a  gain  of  50  per  cent,  and  the  wagon  at  a  loss  of  10  per  cent. 
He  received  for  both  $  195.     What  did  he  pay  for  each  ? 

24.  A  person  engaged  to  reap  a  field  of  grain  for  $  1^ 
per  acre ;  but  leaving  6  acres  not  reaped,  he  received  f  12. 
How  many  acres  of  grain  were  there  ? 


128 


ELEMENTS  OF   ALGEBRA. 


142.    Three  unknown  quantities. 

2x-^    .V  +  3;2=13] 
3x-^2y  -{-    z=z  \0  )■  to  find  x,  y,  and  z. 
2:2=13  I 


1.    Given  < 


x-^3y-\-\ 


J 


Solution. 

2x  +  2/  +  3«  =  13 

(1) 

3x  +  2!/  +  z  =  10 

(2) 

x  +  Zy  +  2z  =  \Z 

(3) 

(2)  X  3, 

9s6  +  6j^  +  32  =  30 

(4) 

(1), 

2x+ 2^  +  3^  =  13 

(4)-(l), 

7x  +  5y==n 

(5) 

(2)X2, 

6x  +  4^  +  2a  =  20 

(6) 

(3), 

x  +  3y  +  2z  =  13 

(6) -(3), 

bx  +  y-l 

(7) 

(7)x5, 
(5). 

25  X  +  5  8*  =  35 

7x+5j/  =  17 

(8) 

(8) -(5). 

18x  =  18 

(9) 

x  =  l 

(10) 

Substituting  (10)  in  (7), 

5  +  ?y  =  7 

(11) 

y  =  2 

(12) 

Substituting  (10)  and  (12) 

in  (l),2  +  2  +  3«  =  13 

(13) 

z  =  Z 

(14) 

The  student  will  observe  that  by  combining  two  of  the  given  equa- 
tions one  of  the  unknown  quantities  is  eliminated,  and  that  by  com- 
bining another  pair  of  the  given  equations  the  same  quantity  is 
eliminated.  We  have  thus  two  equations '  containing  two  unknown 
quantities  which  may  be  readily  solved. 


Find  the  value  of  each  unknown  quantity  in  the  following: 


i»  +    y  -\-    25  =  6  ^ 
x-\-2y  -^^    z  =  7 
x-{-    y-\-2z  =  9 


2x-{-2y+    2;  =  15" 
3x-h    y  +  2z  =  23 
x-3y-^2z  =  ll 


EQUATIONS. 


129 


{     x-{-2y-{-z  =  16] 

4:.  ^  4:X  —  oy  —  z=    GJ> 

[2x-\-2  y-z=  11  ] 

f5x-    y^2z  =  3S^ 
'  2x-\-    y  —    z=    4:  y 
a;  —  32/4-52;  =  44  J 

^-\-    y-\-    z  =  lo 

6.  ^  2x  +  2y—    z=    6 
X—     y-\-4:Z  =  37) 

r     x  +  2y-    z=    6"j 

7.  ^  4.X-    y-{-2z  =  27  [> 
2x-\-2y-{-    2;  =  3lJ 

8.  ^  5a;4-    2/— -"^2;=     14 
[4a;+22/-42;=-  4 

a;—    1/  -4-    2;=  17 

9.  <J      a;-f  4^/  — 62;=    4 
3aj-22^-32;=    7 

-2aj+22/-f-2;=   9  "^ 
5ic  — 3?/=    7  )> 
4^/+    2  =  31J 
a;  4-    y—14: 
2y-Sz=    9 
5  a;  —    z  =   5  j 

2x-  2/  =  9] 
x-22J  =  3  [. 
i/-2^  =  5j 
milne's  el.  of  alg. — 9. 


4: 


10.  { 


11 


12 


13. 


14. 


15.  < 


0,^-12?/ 4-22;= -20  1 

2a;—     y—    z=       7  V 
X-}-   z=     19  J 

6x-  y-\-  z  =  2S 
5x-\-2y-2z  =  12 
2x  +  2y-    z=    8 


4      8 

-4-^  =  5 
4      10 


y^  I 


14 


r  X 


+?+^=    9l 


16.^^  +  1+^=10^ 


a^  +  .V  +  |=19 


(  x-\-y=   9 

17.  ^  a; +2;  =  12 
U  +  ^  =  15. 


18. 


19. 


2/  —  a;-f2;  =  —    5 
z  —  y  —  x  =  —  25 

^*  +  2/  +  ^  =      ^^ 

2aj-32/=        1 

3y  —  4:Z=        7 

[42-5a;  =  -32 


130 


ELEMENTS  OF   ALGEBRA. 


20. 


f  -  4-  -----  ^ 

X      y      z      3 

X      y      z      3  ' 

1+1-1=0 

^  y      z      X         J 

X      y          5 

3_4_      1 

^  y      z         5 

► 

3     4__1 

.21      a;          2  . 

2x-    I 

^  = 

G 

22.     < 

x  +  y  —  2z  = 

—  2 

4aj  — 32/+  z  = 

11 

'  X        z        o  ' 
2      15  ~ 

23.     < 

5       5 

U      6         J 

> 

21. 


25a; -201/ +  152:=     80" 
24.     ^  15a; -252/ +  202;=      <50 
20a; +  152/ -25:2  =  -40 

25.  A  farmer  has  sheep  in  three  pastures.  The  number  in 
the  first  plus  \  of  the  number  in  the  second  plus  \  of  the 
number  in  the  third  equals  70.  The  number  in  the  first 
plus  \  of  the  number  in  the  second  minus  \  of  the  number 
in  the  third  equals  30.  The  number  in  the  second  plus  ^  of 
the  number  in  the  first  plus  y^  of  the  number  in  the  third 
equals  61.     How  many  sheep  are  there  in  each  pasture? 

26.  Henry,  James,  and  Ralph  together  have  50  cents. 
Henry's  and  Ralph's  money  amounts  to  35  cents ;  James' 
and  Ralph's  amounts  to  40  cents.  How  many  cents  has 
each? 

27.  A  farmer  has  wheat,  oats,  and  barley.  The  number 
of  bushels  of  wheat  and  oats  is  200  ;  the  number  of  bushels 
of  wheat  and  barley  is  190 ;  and  the  number  of  bushels  of 
oats  and  barley  is  90.  How  many  bushels  are  there  of  each 
kind  of  grain? 


INVOLUTION. 


143.  1.  What  is  a  power  of  a  quantity ?     An  exponent? 

2.  How  many  times  is  a  quantity  used  as  a  factor  in 
producing  the  second  power?  The  third  power?  The 
fourth  power?     The  fifth  power?     Any  power? 

3.  What  sign  has  the  second  power  of  -|-a?  The  third 
power?  The  fourth  power?  Any  power  of  a  positive 
quantity  ? 

4.  What  sign  has  the  second  power  of  —  a?  The  third 
power?  The  fourth  power?  The  fifth  power ?  The  sixth 
power?  Any  even  power  of  a  negative  quantity?  Any 
odd  power  of  a  negative  quantity  ? 

5.  Which  powers  of  a  negative  quantity  are  positive? 
Which  are  negative  ? 

6.  What  is  the  third  power  of  a^?  The  fourth  power? 
The  fifth  power?     The  sixth  power?    The  nth  power? 

7.  What  is  the  third  power  of  a^?  The  fourth  power? 
The  fifth  power  ?     The  sixth  power  ?     The  nth  power  ? 

8.  How  is  the  exponent  of  a  power  of  a  quantity  deter- 
mined from  the  exponent  of  the  quantity  ? 

144.  The  process  of  finding  a  power  of  a  quantity  is 
called  Involution. 

131 


182 


ELEMENTS   OF   ALGEBRA. 


146.  Principles.  1.  All  poivers  of  a  jjositive  quantitr/ are 
positive. 

2.  All  even  poicers  of  a  negative  quantity  are  positive,  and 
all  odd  powers  are  negative. 

3.  The  exponent  of  the  power  of  a  quantity  is  equal  to  the 
exponent  of  the  quantity  muUiplied  by  the  exponent  of  the 
power  to  ivhich  the  quantity  is  to  be  raised. 

146.    Involution  of  monomials. 

1.    What  is  the  fourth  power  of  3a^ar^? 

Solution.     (3  a'^x^y  =  3  a-x^  •  3  a'^x^  •  3  a'^x^  •  3  a%3  =:  81  a^x^'^. 

Rule.  Raise  the  numerical  coefficient  to  the  required 
power;  multiply  the  exponent  of  each  literal  quantity  by  the 
exponent  of  the  p)ower  to  which  it  is  to  be  raised,  and  prefix 
the  proper  sign  to  the  residt. 


2.    What  is  the  third  power  of 


2a&% 


Solution.      ( ^ab^V  =  ^ah^  ^  2al^  ^  2a^  ^  i 
V3r.V/        SxV     3x'V     Sx'Y      2 


27  x^y^' 


In  raising  a  fraction  to  a  power,  both  numerator  and  denominator 
must  be  raised  to  the  required  power. 

Mnd  the  values  of  the  following : 


3. 

(2a^yy. 

10. 

{lox^y 

17. 

{-2aj'by. 

4. 

{Axyzy. 

11. 

(Sa'bcy. 

18. 

{-lOa^by. 

5. 

{-Sa^zy. 

12. 

(-2m'ny. 

19. 

{a^bh'dy. 

6. 

{2a%y. 

13. 

(-4  amy. 

20. 

(-a^b'my. 

7. 

(-Aab^xy. 

14. 

(2  ab'dy. 

21. 

(-2a'b'dey. 

8. 

(-sc'd'y. 

15. 

(-6a'my. 

22. 

{-x^fzy. 

9. 

{-^yzy. 

16. 

{2x'yy. 

23. 

{x^y^zy. 

INVOLUTION.  133 


24.    (:2xY^T.  32.    f_J^y^].       37. 


(■ 


25.  (-2&2c-d2")^              V     2amV  V^'^' 

26.  (a-^6^c^)l  33     ra'bW  ^^     /8  g^ftV 

27.  (5a'%V)^               '    V^"/^>'*  *    V^yj' 

28.  (_aV"2r*^")-l  3^     /5_arxy\\  3^     /aVV". 


29.  {2a'hxy)\  '    V^mn'zJ  '    \byj 

30.  f^^]\  35.    r^^V.  40.    (-.,    -'y 


3mxJ  \xYzy  \     10  a'm 

31.    r^^Y.  36.    r^^^Y.  41.    f  ^''^' 


4aVy  V4m^nW  '    \mVaf 


147.    Involution  of  polynomials. 

(a  +  &)-  =  (a  +  6)  (a  +  ^)  =  a-  +  2  a&  +  ft^. 

(a  -  by  =  {a  -h)  {a  -  h)=  or  -2  ah  -\-  h\ 

(a  +  by  =  {a  +  &)  (a  +  &)  (a  +  6)  =  d' -\-3o?b  +  Saft^  +  6\ 

(a  _  bf  =  (a-b){a-  b)  (a  -b)  =  a^-^  Sa'b  +  3ab^  -  b\ 

(a  -  bY=-a'  -  4.a'b  +  6a'b'  -  4a6'^  +  b\ 

By  examining  carefully  the  letters,  the  exponents,  the  coefficients, 
and  the  signs  of  the  above  powers,  the  student  will  be  able  to  formu- 
late laws  for  writing  out  the  powers  of  quantities. 


15.  (a' -{-by. 

16.  (a  +  &  +  c)l 

17.  (a-^b  —  cy, 

18.  (x-y-zy. 

19.  {2xi-2y  —  zy. 

20.  (a"  +  ^>"  +  c")-. 

21.  (a"^-6"^-c2-)2. 


Raise  to  the  required 

power : 

1. 

(x-yy. 

8. 

{7n-3ny. 

2. 

{3a-\-2by, 

9. 

{x  +  2yy. 

3. 

(5a  +  4c)2. 

10. 

{2a-oby. 

•4. 

(x-i-yy. 

11. 

{3x  +  2yy. 

5. 

{X  -  yy. 

12. 

{Ax~3yy. 

6. 

{a  +  by. 

13. 

(a  +  by. 

7. 

(x-yy. 

14. 

(x  +  iy. 

EVOLUTION. 


148.  1.  What  are  the  tmo  equal  factors  of  25  ?  Of  36? 
Of  64? 

2.  What  is  one  of  the  two  equal  factors,  or  what  is  the 
second  or  square  root  of  25  ?     Of  81  ?     Of  100  ? 

3.  What  is  one  of  the  three  equal  factors,  or  what  is  the 
third  or  cube  toot  of  a"  ?     Of  8  a^  ?    Of  27  a^?    Of  8  a«  ? 

4.  What  is  the  sign  of  any  power  of  a  positive  quantity  ? 
Siiice  any  power  of  a  positive  quantity  is  positive,  what 
sign  may  one  of  the  equal  factors,  or  a  root  of  a  positive 
quantity  have  ? 

5.  What  is  the  sign  of  the  second  power  of  a  negative 
quantity  ?  The  fourth  power  ?  The  sixth  power  ?  Any 
even  power  ? 

6.  Since  any  even  power  of  a  positive  or  negative  quan- 
tity has  the  positive  sign,  what  is  the  sign  of  the  everi  root 
of  any  quantity  ? 

7.  What  sign  has  the  second  power  of  —  3  ?  The  third 
power?  The  fourth  power?  The  fifth  power?  What 
powers  of  a  negative  quantity  are  negative  ? 

8.  Since  a  power  having  the  negative  sign  is  the  product 
of  an  odd  number  of  equal  negative  factors,  what  is  the 
sign  of  the  odd  root  of  a  negative  quantity  ? 

134 


EVOLUTION.  135 

9.  What  kind  of  quantity  used  twice  as  a  factor  will 
give  a  product  with  the  negative  sign?  What  kind,  used 
four  times ?     Any  even  number  of  times? 

10.  Since  no  quantity  used  an  even  number  of  times  as 
a  factor  will  give  a  negative  product,  what  may  be  said 
regarding  an  even  root  of  a  negative  quantity  ? 

149.  One  of  the  equal  factors  of  a  quantity  is  called  the 
Root  of  the  quantity. 

Thus,  3  is  a  root  of  9,  of  27,  etc. ;  a  is  a  root  of  a^,  of  a*,  etc. 

150.  Roots  are  named  from  the  number  of  equal  factors 

into  which  the  quantity  is  resolved  or  separated. 

Thus,  one  of  the  two  equal  factors  is  called  the  second  or  square 
root ;  one  of  the  three  equal  factors,  the  third  or  c^ihe  root ;  one  of  the 
four  equal  factors,  the,  fourth  root,  etc. 

151.  The  root  of  a  quantity  is  indicated  by  placing  the 
sign  -yj,  called  the  Radical  Sign,  before  the  quantity. 

A  number  or  quantity  called  the  Index  is  frequently 
written  at  the  opening  of  the  radical  sign,  to  show  what 
root  is  sought. 

Thus,  y/a  shows  that  the  third  root  of  a  is  sought ;  V«,  the  fifth 
root,  etc. 

When  no  index  is  w^ritten  in  the  opening  of  the  radical 
sign,  the  second  or  square  root  is  indicated. 

152.  The  process  of  finding  the  root  of  a  quantity  is 
called  Evolution. 

153.  Principles.  1.  An  odd  root  of  a  quantity  has  the 
same  sign  as  the  quantity  itself. 

2.  An  even  root  of  a  positive  quantity  is  either  positive  or 
negative, 

3.  An  even  root  of  a  negative  quantity  is  impossible  or 
imaginary. 


136  ELEMENTS   OF   ALGEBRA. 

154.    Evolution  of  a  monomiaL 

1.    What  is  the  square  root  of  16a^x^? 

PROCESS  Explanation.       Since   in   squaring    a 

monomial  we  square  the  coefficient  and 

a/16  aV  =  +  4  a^x^       multiply  the  exponents  of  the  letters  by 

2,  to   extract  the   square   root   we    must 

extract  the  square  root  of  the  coefficient,  and  divide  the  exponents  of 

the  letters  by  2. 

The  sign  of  the  root  is  either  plus  or  minus  (Prin.  2).  Hence  the 
square  root  of  the  quantity  is  ±  4  a^x^. 

Rule.  Extract  the  required  root  of  the  numerical  coeffi- 
cient;  divide  the  exponent  of  each  letter  by  the  index  of  the 
root  sought ;  and  prefix  the  proper  sign  to  the  result. 

The  root  of  a  fraction  is  found  by  taking  the  root  of  the  numerator 
and  of  the  denominator  separately. 

Find  the  values  of  the  following : 


2.    V4a26l  13.    -Vx'^fz'^ 


3.  ^8Sy.  14.    V-64a^Y^'^ 

4.  V25¥y\  ^''    ViOOW^ 

6.    V--27m'V.  

, 17.    ^x'^y'^z'''. 

6.    </l6a'b'c''. 


18.     Va^'b'^c''^ 


7.  iZ-a'^b'^c^'. 

8.  ^-125lcV^. 


19.    V_32m^Va:25^<. 
20. 


9.    -Va^W^dy^ 


3      /i^'. 
\9xY 


10.  V49m«n^".  21.    J\ 

11.  v'Sl  aWe\ 


25m*n^ 


36  0"%^ 


22 


12.    ■</^S2aVy'^,  \27x'y 


\27a;VV^ 


EVOLUTION.  137 


23.    J-Jl^^,  25.    Jll^YL. 

\      343  a;  V'  yllUm'^n'^ 


24       ^ja^y^  2^       W^GoW)^ 

155.    To  extract  the  square  root  of  a  polynomial. 

1.  Since  a^ -\- 2  ab -\- b^  is  the  square  of  (a  +  6),  what  is 
the  square  root  of  a^  +  2  a6  +  6^? 

2.  How  may  the  first  term  of  the  square  root  be  found 
from  the  lirst  term  of  a^  +  2  a6  +  6^? 

3.  Since  the  first  term  of  the  root  is  found  to  be  a,  how 
may  the  second  term  be  found  from  2ab,  the  second  term 
of  the  power  ? 

1.    Find  the  square  root  of  a^  +  2  a6  +  61 

PROCESS. 

a^  +  2  a6  +  6M  a  4-  6 


Trial  divisor         =        2  a 
Complete  divisor =2  a  -f  & 


2  a6  +  6^ 
2  ab  +  b^ 


Explanation.  The  quantities  are  arranged  according  to  the 
descending  powers  of  a.  Since  the  first  term  of  the  quantity  is  a^^ 
its  square  root  a  is  the  first  term  of  the  root  sought.  Subtracting  a^ 
from  the  quantity,  there  is  a  remainder  of  2  ab  -\-  b'^. 

Since  the  first  term  of  the  remainder  is  2  ab^  if  it  is  divided  by  2  «, 
twice  the  root  already  found,  the  quotient  b  will  be  the  second  term 
of  the  root.  2  a,  or  twice  the  root  found,  is  called  the  t7'ial  divisor. 
The  complete  divisor^  or  the  divisor  which  multiplied  by  b  will  pro- 
duce the  remainder  2ab  -\-  b^,  is,  therefore,  2a  +  b.  It  is  formed  by 
adding  to  the  trial  divisor  the  second  term  of  the  root.  Multiplying 
the  complete  divisor  by  b  and  subtracting,  there  is  no  remainder. 
Therefore  a  +  6  is  the  square  root  of  the  quantity. 


138  ELEMENTS   OF  ALGEBRA. 

2.    Find  the  squarjB  root  of  x^  —  6x^-{- 13  x^  —  12  a;  +  4. 

PROCESS. 


2  aj2  -  3  X 


-6x^  +  13x^ 
-6x^-^    9x^ 


2x^-6x  +  2 


4aj2_  12  0^  +  1: 


Explanation.  The  process  is  the  same  as  in  the  previous  exam- 
ple until  the  first  two  terms  of  the  root  are  found.  The  first  two 
terms  are  thus  found  to  be  x^  —  3  x. 

To  find  the  next  term  of  the  root  we  consider  x^  —  3  x  as  one  quan- 
tity, which  we  multiply  by  2  for  the  trial  divisor.  Dividing  the  first 
term  of  the  remainder  by  the  first  term  of  this  trial  divisor,  the  third 
term  of  the  root  is  obtained,  which  is  4-  2.  Annexing  2  with  its 
proper  sign  to  the  trial  divisor  already  found,  the  complete  divisor  is 
obtained  which  is  2  x^  —  6  x  +  2.  This  when  multiplied  by  2  gives  as 
a  product  ix'^  —  12  x  +  4,  which  subtracted  from  4  x"^  —  12  ic  +  4  leaves 
no  remainder.  Therefore,  the  square  root  of  the  quantity  x*  —  6  x^ 
+  13x2-12x  +  4  isx2-3x  +  2. 

Rule.  Arrange  the  terms  of  the  polynomial  with  refer- 
ence to  the  consecutive  powers  of  some  letter. 

Extract  the  square  ix^ot  of  the  first  term,  write  the  result  as 
the  first  term  of  the  root,  and  subtract  its  square  from  the 
given  polynomial. 

Divide  the  first  term  of  the  remainder  by  twice  the  root 
already  found,  as  a  trial  divisor,  and  the  quotient  will  be  the 
next  term  of  the  root.  Write  this  result  in  the  root,  and  annex 
it  to  the  trial  divisor  to  form  a  complete  divisor. 

Multiply  the  complete  divisor  by  this  term  of  the  root,  and 
subtract  the  product  from  the  first  remainder. 

Continue  in  this  manner  until  all  the  terms  of  the  root  are 
found. 


EVOLUTION.  139 

Find  the  square  root  of  the  following : 

3.  x^ -{- 4:xy -\- iy^. 

4.  a--6a4-9. 

5.  16aj-  +  24a;?/-f  9/. 

6.  9x'-12xy-^4y\ 

7.  25m^-{-mm-\-36. 

8.  100a- -20  a  4-1. 

9.  81a2  4-90a  +  25. 

10.  36x''-^96xy  +  6iy'. 

11.  a'-^b'-}-c'-\-2ab~2ac-2bc. 

12.  aj2  +  2a;?/4-8x  +  r  +  8y+16. 

13.  x''-\-4.y--4.xy-4:XZ-^Az^-\-Syz, 

14.  4a- +  96- -f- -4 -12a&  4- 8a -126. 

15.  4a'-M-4a6-4a  +  6--26  4-l. 

16.  a^-2a'^-3a2  +  4a  +  4. 

17.  4:X^-\-4.x^-}-5x^-^2x-j-l. 

18.  9a2  4-c2-6ac  +  30a-10c  +  25. 

19.  16m^  +  32m3-16m2-32m4-16. 

20.  30a'''  +  39a2  +  25a^4-18aH-9. 

21.  llm-  +  6m^H-m^  +  6m  +  l. 

22.  4:-\-4.x  —  4:y-\-.x'^-{-y-  —  2xy. 

23.  m*  +  6m*^  +  9m^  —  4?/^  —  12m7i 4- 4nl 

24.  4a^-4a26+ 62_^24a2_126  4-36. 

25.  x^-4x^-\-5x'^-2x-{-l. 

26.  a;2  4-2a;2/-2a;;2-2?/;^4-2/^4-;?^ 


140  ELEMENTS  OF   ALGEBRA. 

156.  To  extract  the  square  root  of  numbers. 

1-'=    1  10=^=    100  100-=    10000 

92  =  81  992  =  9801  9992  =  998001 

The  student  should  observe  carefully  the  number  of 
figures  required  to  express  the  square  of  units^  tens,  and 
hundreds  in  the  above  examples. 

The  square  of  the  smallest  and  of  the  largest  number 
of  each  order  is  given,  consequently  the  number  of  figures 
required  to  express  the  square  of  a  number  can  be  readily 
discovered. 

1.  How  many  figures  are  required  to  express  the  square 
of  any  number  of  units  9 

2.  How  does  the  number  of  figures  required  to  express 
the  second  power  of  any  number  between  9  and  100  com- 
pare with  the  number  of  figures  in  the  number  ? 

3.  How  does  the  number  of  figures  expressing  the  second 
power  of  any  number  between  99  and  1000  compare  with 
the  number  of  figures  in  the  number  ? 

4.  If  the  second  power  of  a  number  is  expressed  by  3 
figures,  how  many  orders  of  units  are  there  in  the  number? 

If  by  4,  how  many  ?     By  5  ?     By  7  ? 

157.  Principles.  1.  The  square  of  a  number  is  expressed 
by  twice  as  many  figures  as  is  the  number  itself,  or  by  one  less 
than  twice  as  many. 

2.  The  orders  of  units  in  the  square  root  of  a  number  cor- 
respond to  the  number  of  periods  of  two  figures  each  into 
ivhich  the  number  can  be  separated,  beginning  at  units. 

The  left-hand  period  may  contain  only  one  figui'e. 

158.  If  the  tens  of  a  number  are  represented  by  t,  and  the 
units  by  u,  the  square  of  a  number  consisting  of  tens  and 
units  will  be  the  square  of  {t  +  u)  or  t^  -\-2  tu  -f-  u^. 


2t  =60 


2t^u  =66 


EVOLUTION.  141 

Thus,  35  =  3  tens  plus  5  units,  or  30  +  5,  and  35"^  =  30*^  +  2(30  x  5) 
+  52  =  1225. 

1.  What  is  the  square  root  of  1296  ? 

PROCESS.  Explanation.     According  to  Prin . 

1 2  .  96 1  SO  4-  fi     ^'  ^^^  orders  of  units  in  tlie  square 
•  JD I  t3U  i-  o     j.^^^  ^£  ^  number  may  be  determined 
t  =  900  by  separating  the  number  into  periods 

396  of  two  figures  each,  beginning  at  units. 

Separating  1296  thus,  there  are  found 
to  be  two  orders  of  units  in  the  root ; 
^"^  that  is,  it  is  composed  of  tens  and 

units.  Since  the  square  of  tens  is 
hundreds,  and  the  hundreds  of  the  power  are  less  than  16  or  42,  and 
more  than  9  or  3^,  the  tens'  figure  of  the  root  must  be  3.  3  tens,  or 
30,  squared  is  900,  and  900  subtracted  from  1296  leaves  396,  which  is 
equal  to  2  times  the  tens  multiplied  by  the  units  plus  the  units  squared. 
Since  2  times  the  tens  multiplied  by  the  units  is  much  greater  than 
the  square  of  the  units,  396  is  nearly  2  times  the  tens  multiplied  by 
the  units.  Therefore,  if  396  is  divided  by  2  times  the  tens,  or  60,  the 
quotient  will  be  approximately  the  units  of  the  root.  Dividing  by  60 
the  trial  divisor^  the  units  are  found  to  be  6.  And  since  the  complete 
divisor  is  found  by  adding  the  units  to  twice  the  tens,  the  complete 
divisor  is  60  +  6,  or  QQ.  This  multiplied  by  6  gives  as  a  product  396, 
which,  when  subtracted  from  396,  leaves  no  remainder.  Therefore, 
the  square  root  of  1296  is  36. 

Any  number  may  be  regarded  as  composed  of  tens  and 
units;  hence  the  process  given  above  has  a  general  appli- 
cation. 

Thus,  495  may  be  considered  as  49  tens  and  5  units ;  3843  as  384 
tens  and  3  units. 

2.  Find  the  square  root  of  61009. 

Solution.  6  .  10  •  09  |  247 

4 

Trial  divisor  =      2  x    20  =    40  I  210 

Complete  divisor  =    40  +      4  =    44     176 
Trial  divisor  =      2  x  240  =  480      3409 

Complete  divisor  =  480  +      7  =  487      3409 


142  ELEMENTS  OF  ALGEBRA. 

Rule.  Separate  the  member'  into  periods  of  tico  figures 
each,  heg inning  at  units. 

Find  the  greatest  square  in  the  left-hand  j^eriod^  and  write 
its  root  for  the  first  figure  of  the  required  root. 

Square  this  root,  subtract  the  result  from  the  left-hand 
period,  and  annex  to  the  remainder  the  next  period  for  a  new 
dividend. 

Double  the  root  already  found,  ivith  a  cipher  annexed  for  a 
trial  divisor,  and  by  it  divide  the  dividend.  The  quotient,  or 
quotient  diminished,  ivill  be  the  second  figure  of  the  root.  Add 
to  the  trial  divisor  the  figure  last  founds  multiply  this  complete 
divisor  by  the  figure  of  the  root  found,  subtract  the  product 
from  the  dividend,  and  to  the  remainder  annex  the  next  period 
for  the  next  dividend. 

Proceed  in  this  manner  until  all  the  periods  have  been  used- 
The  result  will  be  the  square  root  sought. 

Decimals  are  pointed  off  into  periods  of  two  figures  each,  by 
beginning  imth  tenths  and  passing  to  the  right. 

Extract  the  square  root  of  the  following : 

3.  729.  12.  9409,  21.  978121. 

4.  6561.  13.  55225.  22.  .004096. 

5.  9604.  14.  26569.  23.  19.1844. 

6.  5329.  15.  13924.  24.  620.01. 

7.  7225.  16.  103041.  25.  .00632025. 

8.  4356.  -  17.  180625.  26.  12.6025. 

9.  2916.  18.  253009.  27.  17.3056. 

10.  2401.       19.  185761.      28.  331.9684. 

11.  9025.       20.  265225.     29.  13.7641. 


EVOLUTION.  143 

159.    To  extract  the  cube  root  of  a  polynomial. 

1.  Since  a^  -\-3a-b  4-  3a&-  +  6"  is  the  third  power  or  cube 
of    (a-\-b),  what   is   the  third  or  cube    root   of   a^-\-3a^b 

•^Sab'  +  b^? 

2.  How  may  the  first  term  of  the  root  be  found  from  the 
first  term  of  the  power,  a^  ? 

3.  Since  the  first  term  of  the  root  is  known  to  be  a,  how 
may  the  second  term  of  the  root  be  found  from  the  second 
term  of  the  power,  3  a^b  ? 

4.  Since  a^ -}- 3 a-b -\- 3 ab'- -{- b^  is  the  cube  of  (a  +  b), 
after  the  cube  of  the  first  term  of  the  root  has  been  sub- 
tracted from  a^  -\-3a^b  -}-  3ab^  +  b%  by  what  quantities  must 
b,  the  second  terra,  be  multiplied  to  make  the  entire  power  ? 

1.    Find  the  cube  root  of  x'  +  3x-y  -\-3xy^  +  y^. 

PROCESS. 

oc^  -\-  3x-y-\-  3xy^  -{-  y^lx  -}-  y 
x" 


Trial  divisor         =3a^ 

Complete  divisor  =  3  x'  -{-  3  xy  -j- y- 


3x'y-^3xy'-\-f 
3x^y-{-  3xy^  -\-  y^ 


Explanation.  Since,  if  the  quantity  is  a  cube,  one  of  the  terms  is 
a  cube,  the  first  term  of  the  root  is  the  cube  root  of  x^^  which  is  x. 
Subtracting  x^  from  the  quantity,  there  is  left  3  x^y  +  3  xy^  +  y^. 
Since  the  second  term  of  the  root  can  be  found  from  the  first  term  of 
the  remainder  by  dividing  it  by  three  times  the  square  of  the  root 
already  found,  the  second  term  of  the  root  of  the  quantity  will  be 
found  by  dividing  3  x-y  by  3  x^,  the  trial  divisor,  which  gives  y  for  the 
second  term  of  the  root.  Since  the  last  three  terms  of  the  cube  of 
a  binominal  are  composed  of  the  product  of  the  second  term  by  three 
times  the  square  of  the  first,  three  times  the  product  of  the  first  multi- 
plied by  the  square  of  the  second  and  the  cube  of  the  second,  3  x^ 
+  Sxy  -\-  y'^  is  the  complete  divisor. 

The  complete  divisor  multiplied  by  y  is  3  x'^y  +  3  xy^  +  y^,  and  this 
subtracted  from  the  given  quantity  leaves  no  remainder.  Therefore, 
xi-y  is  the  cube  root  of  a;^  +  3  xi^y  +  Sxy^  -\-  y^. 


144  ELEMENTS  OF   ALGEBRA.    " 

J  f  (x  -{-  y  -{-z)  is  raised  to  the  third  power,  the  first  two 
terms  may  be  represented  by  a,  or  the  cube  of  (x-\-y-\-z) 
may  be  represented  by  a"  -\-3a-z  -\-3  az^  -\-  z^,  in  which  a 
stands  for  (x-\-y).  Hence  it  is  apparent  that  the  cube 
root  of  a  quantity  whose  root  consists  of  more  than  two 
terms  may  be  found  precisely  as  in  Example  1,  by  consider- 
ing the  terms  already  found  as  one  term, 

2 .  Find  the  cube  root  of  a^  —  3  a^  -|-  6  a^  —  7  a^  +  6  a!' 
-3a-f  1. 

Solution.  a^-S  a^^Qa^-1  a^-^Q  «2_3  a+ 1  ^g^j-gj^i 

3  a* 


3a*-6a3  4.3Qj2 


-3a5+6a*-7rt3 


3a*-6a3  +  6a2_3a+i 


E,uLE.  Arrange  the  polynomial  with  reference  to  the  consec- 
utive poivers  of  some  letter. 

Extract  the  cube  root  of  the  first  term,  write  the  result  as  the 
first  term  of  the  root,  and  subtract  its  cube  from  the  given 
polynomial. 

Divide  the  first  term  of  the  remainder  by  three  times  the 
square  of  the  root  already  found,  as  a  trial  divisor,  and  the 
quotient  will  be  the  next  term  of  the  root. 

Add  to  this  trial  divisor  three  times  the  product  of  the  first 
and  second  terms  of  the  root,  and  the  square  of  the  second 
term.     The  result  will  be  the  complete  divisor. 

Multiply  the  complete  divisor  by  the  last  term  of  the  root 
found,  and  subtract  this  product  from  the  dividend.  Con- 
tinue in  this  manner  until  all  the  terms  of  the  root  are  found. 

Find  the  cube  root  of  the  following : 

3.  a^-\-?>a%-\-3aW^b\  5.    a^  +  6a;2  4-12a; +  8. 

4.  x^'-^x^y  +  ^xy^^f,  6.   21  a^ +  21  a^ +  ^a^-l. 


EVOLUTION.  145 

7.  2lQ(?-21a^y-Jt^xy''-f. 

8.  x^^l2x^-\-i:%x'-^^L 

9.  ^m^  — 12  m'^n-{-Qmn^  —  n\ 

10.  x^-{-^x^-[-^x^^:x^. 

1 1 .  a?m^  +  6  a^m^^  + 12  amft^  +  8  h\ 

12 .  8  a^'  -  60  aV  + 150  aV  -  125  c\ 

13.  27  +  54&  +  3662  4.8^^ 

14.  l  +  Sa  +  Sa^  +  a^. 

15.  m«  +  18  m*  +  108  m^  +  216. 

16.  8.r^  +  36a;V  +  54a;^2^  272/3^ 

17 .  64  0^9  +  576  it-V  +  1728  :x^y'  +  1728  y\ 

18.  ft;'''-3aa^4-5aV-3a^aj-a«. 

19.  8.^*^-84  052  4- 294 a; -343. 

20.  343a^  +  588a;22/  +  336ajz/2  +  642/^ 

21.  a^  +  9  a«63  +  27  «'W' +  27  &^. 

22 .  8  a;«  4-  72  xY  +  216  o^y  +  216  y\ 

23 .  a^^^  -  9  a'W  +  27  a-6  -  27. 

24.  l  +  9a5  +  27a;2_^27ar^. 

25.  m^2  —  3  mhi  +  3  mhi^  —  nl 

26.  a;«-6.^'^'  +  15a;^-20a^  +  15a;2-6a;  +  l. 

27.  a«-9a^  +  33a^-63a'^  +  66a--36a  +  8. 

28.  a;6-3a^  +  5a;3_3^_-j^ 

29.  27  m«  -  135  m^  +  171  m*  +  55  rrv"  -  114  m^  _  60  m  -  8. 

30.  8a;«  +  36 aj^.y  +  42  xY^  -^o^rf-21  ci^y*  +  9 xy' - y\ 

31.  1  -  9.T  +  39ic2  _  99^3  ^  -(^55^4  _  -^44^5  ^  ^4^6 

32.  1  +  12aj  +  60  0^  +  UOx^  -f-  240x^  +  192  o^  +  Ux\ 

milne's  el.  of  alg. — 10. 


146  ELEMENTS   OF   ALGEBRA. 

160.  To  extract  the  cube  root  of  numbers. 

13=      1  10'3=      1000  100^=      1000000 

3^=    27  36^=    4:6656  361^=    47044881 

9^  =  729  99^=970299  999^  =  997002999 

The  student  should  observe  carefully  the  number  of 
figures  required  to  express  the  cube  of  units,  tens,  and 
hundreds  in  the  above  examples. 

The  cube  of  the  smallest,  of  the  largest,  and  of  an  inter- 
mediate number  of  each  order  is  given,  consequently  the 
number  of  figures  required  to  express  the  cube  of  any  num- 
ber may  be  readily  discovered. 

1.  How  many  figures  are  required  to  express  the  cube 
of  any  number  of  units? 

2.  How  does  the  number  of  figures  required  to  express 
the  cube  of  any  number  between  9  and  100  compare  with 
the  number  of  figures  expressing  the  number  ? 

3.  How  does  the  number  of  figures  required  to  express 
the  cube  of  any  number  between  99  and  1000  compare  with 
the  number  of  figures  expressing  the  number  ? 

4.  If,  then,  the  cube  of  a  number  is  expressed  by  4 
figures,  how  many  orders  of  units  are  there  in  the  root  ? 
If  by  5  figures,  how  many  ?  If  by  6  figures,  how  many  ? 
If  by  8  figures,  how  many  ? 

5.  How  may  the  number  of  figures  in  the  cube  root  of  a 
number  be  found  ? 

161.  Principles.  1.  The  cube  of  a  number  is  expressed 
by  three  times  as  many  figures  as  the  number  itself  or  by  one 
or  two  less  than  three  times  as  many. 

2.  The  orders  of  units  in  the  cube  root  of  a  number  corre- 
spond to  the  number  of  periods  of  three  figures  each  into  which 
the  number  can  be  separated,  beginning  at  units. 

The  left-hand  period  may  contain  one,  two,  or  three  figures. 


EVOLUTION 


147 


162.  If  the  tens  of  a  number  are  represented  by  t,  and 
the  units  by  u,  the  cube  of  a  number  consisting  of  tens  and 
units  will  be  the  cube  of  {t  +  u)  or  ^  +  3^-i^  +  (^tiv^  +  u^. 

Thus,  35  =  3  tens  +  5  units,  or  30  +  5,  and  35^  =  30^  +  3  (302  x  5) 
+  3(30  X  52) +53  =^42875. 

1.    What  is  the  cube  root  of  74088  ? 


Trial  divisor 


Complete  divisor 


PROCESS. 

74.088  I  40 +  2 

^=  64  000 

3^2^4800 
^tu=     240 

?r=        4 


=  5044    10088 


10088 


Explanation.  According  to  Prin.  2,  the  orders  of  units  may  be 
determined  by  separating  the  number  into  periods  of  three  figures 
each,  beginning  at  units.  Separating  74088  thus,  there  are  found  to 
be  two  orders  of  units  in  the  root ;  that  is,  it  is  composed  of  tens 
and  units.  Since  the  cube  of  tens  is  thousands,  and  the  thousands 
of  the  power  are  less  than  125  or  5-^  and  more  than  64  or  4^,  the  tens' 
figure  of  the  root  must  be  4.  4  tens,  or  40,  cubed  is  64000,  and  64000 
subtracted  from  74088  leaves  10088,  which  is  equal  to  three  times  the 
tens2  X  the  units  +  3  times  the  tens  x  the  units2  +  the  units^. 

Since  three  times  the  tens2  is  much  greater  than  three  times  the 
tens  multiplied  by  the  square  of  the  units,  10088  is  a  little  more  than 
three  times  the  tens  squared  multiplied  by  the  units.  If,  then,  10088  is 
divided  by  3  times  the  square  of  the  tens,  or  4800,  the  trial  divisor^ 
the  quotient  2  will  be  the  units  of  the  root,  provided  proper  allowance 
has  been  made  for  the  additions  necessary  to  obtain  the  complete 
divisor.  4800  is  contained  in  10088  twice  ;  consequently  the  second 
figure  of  the  root  or  the  units  is  2. 

Since  the  complete  divisor  is  found  by  adding  to  three  times  the 
square  of  the  tens,  three  times  the  tens  multiplied  by  the  units,  and 
the  square  of  the  units,  the  complete  divisor  will  be  4800  +  240  +  4  or 
5044.  This  multiplied  by  2  gives  as  a  product  10088,  which,  sub- 
tracted from  10088,  leaves  no  remainder.  Therefore,  the  cube  root  of 
74088  is  42. 


148  ELEMENTS   OF   ALGEBRA. 

Since  any  number  may  be  regarded  as  composed  of  tens 
and  units,  the  process  given  above  has  a  general  application. 

2.    What  is  the  cube  root  of  1860867  ? 


SOLUTION. 


Trial  divisor  =   3(10)2  =  300 
3(10  X  2)  =    60 

22  Z3      4 


Complete  divisor 

= 

=  364 

Trial  divisor  =  3(120)2  = 

z  43200 

3(120  X 

3)  = 

=    1080 

32  z 

9 

1.860-867  IJ23 

1 

860 


728 


Complete  divisor  =  44289 


132867 


132867 


EuLE.  Separate  the  nitmher  into  periods  of  three  figures 
each^  beginning  at  units. 

Find  the  greatest  cube  in  the  left-hand  period,  and  ivrite  its 
root  for  the  first  figure  of  the  required  root.  Cube  the  root, 
subtract  the  result  from  the  left-hand  period,  and  annex  to  the 
remainder  the  next  period  for  a  dividend. 

Take  three  times  the  square  of  the  root  already  found  ivith 
a  cipher  annexed,  for  a  trial  divisor,  and  by  it  divide  the 
dividend.  The  quotient  or  quotient  diminished  will  be  the 
second  figure  of  the  root.  To  this  trial  divisor  add  three  times 
the  product  of  the  first  part  of  the  root  with  a  cipher  annexed, 
multiplied  by  the  second  part,  and  also  the  square  of  the  second 
part.     Their  sum  will  be  the  complete  divisor. 

Multiply  the  complete  divisor  by  the  second  part  of  the  root, 
and  subtract  the  product  from  the  dividend.  Continue  thus 
until  all  the  figures  of  the  root  have  been  found. 

Decimals  are  pointed  off  into  periods  of  three  figures  each,  by 
beginning  at  tenths  and  passing  to  the  right. 


EVOLUTION. 


149 


3.   What  is  the  cube  root  of  95256152263? 


Solution. 

95-256. 152.  263 1 4567 

64 

4800  31256 

60C 

25 

5425 

27125 

4131152 

607500 

8100 

36 

615636 

3693816 

437336263 

62380800 

95760 

49 

62476609 

437.336263 

An  abridged  method  of  extracting  the  cube  root  of  a  number  is  pre- 
sented on  page  339  of  the  author's  Standard  Arithmetic. 


Extract  the  cube  root  of  the  following : 


4.  571787. 

5.  148877. 

6.  262144. 

7.  250047. 

8.  704969. 

9.  912673. 

10.  314432. 

11.  614125. 


12.  16581375. 

13.  44361864. 

14.  100544625. 

15.  5545233. 

16.  34965783. 

17.  41063625. 

18.  68417929. 

19.  743677416. 


20.  .015625. 

21.  43614208. 

22.  13312053. 

23.  .004019679. 

24.  .000166375. 

25.  28.094464. 

26.  130323.843. 

27.  48.228544. 


QUADRATIC     EQUATIONS. 


PURE  QUADRATIC  EQUATIONS. 

163.  An  equation  of  the  second  degree  is  called  a  Quad- 
ratic Equation. 

164.  An  equation  which  contains  only  the  second  power 
of  the  unknown  quantity  is  called  a  Pure  Quadratic  Equation. 

165.  A  value  of  the  unknown  quantity  in  an  equation 
above  the  first  degree  is  called*  a  Root  of  the  equation. 

1.  Given  2  x^  —  5  =  45,  to  find  the  value  of  x. 

Solution.  2  x^  —  5  :=  45 

2  x-2  =  45  +  5 
2  x2  =  50 
x-2  =  25 
Extracting  the  square  root,  x  =  ±b 

2.  Given  ax'  -|-  c  =  &x-  +  c?,  to  find  the  value  of  x. 

Solution.  ax^  +  c  =  hx"^  -\-  d 

ax?-  —  bx'^  =  d  —  c 
(a  -  b)x^  =  d-  c 

2  —  d  —  C 

a  —  b 


Extracting  tlie  square  root^         x=  ±  \/— ^ 

^a  ■ 


150 


QUADRATIC  EQUATIONS.  151 

Find  the  values  of  x  in  the  following  equations : 

3.  x^-4.  =  12.  12.    15x^-6  =  7  x^  +  194.. 

4.  a^  +  l  =  10.  13.    x(x-4.)  =  -x^-4:X-\-8. 

5.  2x'-5  =  x'-\-20.  14^    a?-2^    12 

6.  3x'-15  =  57  +  x^.  ^         ^  +  ^ 


X' 


15.    2a:2_8  =  -4-7. 


7.    a:2  4.i^:^_{_4,  3 

4 


8.    5a;2-2  =  2a;'^  +  25. 


16.    ir(2a^+3)=.'K2^3a;-f  81. 


19.    -=4a. 


9.    a^4-18  =  22-3a;2^  ^^-    — 3— +  ^  =  42. 

10.  3a:2_29  =  -  +  510.  ^^^    ^'-9  =  cr-6a. 

4 

11.  8aj2-f  2  =  42-2a;-.  "'"    a 

20.    x(x  —  2a)  =  2xr  —  2ax  —  a^. 

PROBLEMS. 

166.  1.  If  to  four  times  the  square  of  a  certain  number 
3  is  added,  the  sum  is  19.     What  is  the  number  ?    . 

2.  Two  numbers  are  to  each  other  as  2  to  5,  and  the  sum 
of  their  squares  is  261.     What  are  the  numbers  ? 

Suggestion.     Let  2  x  and  5  x  represent  the  numbers. 

3.  If  a  certain  number  is  increased  by  1  and  also 
diminished  by  1,  the  product  of  the  sum  and  difference  is 
48.     What  is  the  number  ? 

4.  The  area  of  one  square  field  is  twice  that  of  another, 
and  both  together  contain  867  square  rods.  What  is  the 
length  of  a  side  of  the  smaller  ? 

Suggestion.    Let  x  represent  length  and  x^  area  of  smaller  field. 

5.  A  lady  paid  f  36  for  a  certain  number  of  yards  of 
cloth.  She  paid  i  as  many  dollars  per  yard  as  there  were 
yards  in  the  piece.     How  many  yards  were  there  ? 


152  ELEMENTS  OF   ALGEBRA. 

6.  What  number  is  that  whose  square  plus  18  is  equal 
to  half  its  square  plus  30|-  ?  * 

7.  There  is  a  rectangular  field  whose  breadth  is  f  of  its 
length.  After  laying  out  -^^  of  the  field  for  a  garden,  there 
remain  54  square  rods.  Find  the  length  and  the  breadth 
of  the  field. 

8.  Two  young  men  were  conversing  about  their  ages. 
The  younger  said,  ^^I  am  25  years  of  age."  "Then/'  said 
the  older,  "  the  sum  of  our  ages  multiplied  by  the  difference 
equals  51."     What  was  the  age  of  the  older  man  ? 

AFFECTED   QUADRATIC   EQUATIONS. 

167.  1.  What  is  the  square  oi  x  +  1?  Oix-{-2?  Of 
x  +  3? 

2.  How  may  the  first  term  of  a  binomial  be  found  from 
its  square  ? 

3.  Since  the  second  term  of  the  square  of  a  binomial  is 
twice  the  product  of  both  terms,  how  may  the  second  term 
of  the  binomial  be  found  when  the  first  term  of  the  binomial 
is  known  ? 

4.  Add  a  quantity  to  x^'\-2x  that  will  make  it  a  perfect 
square.     How  is  the  term  found  ? 

6.  Add  a  quantity  to  o;^  +  10  a;  that  will  make  it  a  perfect 
square.     How  is  the  term  found  ? 

168.  An  equation  which  contains  both  the  first  and 
second  powers  of  an  unknown  quantity  is  called  an 
Affected  Quadratic  Equation. 

Thus,  x2  -f  2x  =  8,  8x2  4-  6ic  =  16,  and  ax'^  -\- bx  =  c  are  affected 
quadratic  equations. 


QUADRATIC   EQUATIONS.  153 

169.    First  method  of  completing  the  square. 

1.  Given  x^-\-8x=  20,  to  find  the  value  of  x. 

PROCESS. 

aj2  + 8.^^  +  16  =  20 -hie 
a^  4- 8a; +  16  =  36 
a;  +  4  =  ±6 

x  =  6  —  4:  or  —6  —  4 
x  =  2  or  -10. 

Explanation.  The  first  member  of  the  equation  will  be  made  a 
perfect  square  by  adding  the  square  of  one  half  the  coefficient  of  x 
to  both  members.  One  half  of  8  is  4  and  the  square  of  4  is  16.  This 
added  to  each  member  gives  a^  -|-  8  x  -f  16  =  36.  Extracting  the  square 
root  of  each  member,  we  have  iTH-  4  =  ±  6.  Using  the  first  value 
of  X,  x  =  2  ;  using  the  second  value  of  x,  x  =  —  10. 

2.  Given  aj^  -j-  3ic  =  28,  to  find  the  value  of  x. 

Solution.  x^  +  .3  x  =  28 

Completing  the  square,  x^  +  3  x  +  f  =  28  +  f  or  if  i- 

Extracting  the  square  root,  x  +  i  =  ±  V- 

Transposing,  x  =  |  or  4 

X  =  -  V-  0] 

3.  Given  3  a;-  -f  4  a;  =  20,  to  find  the  value  of  x. 

Solution.  '  3  x2  +  4  x  =  20 

Dividing  by  the  coefficient  of  x'-^,         x^  +  |  x  =  2^*^- 
Completing  the  square,  x'^  +  |  x  +  |  =  2^0  +  *  or  -^^ 

Extracting  the  square  root,  x  +  f  =  ±  f 

.-.  x  =  2  or  --V>- 


154  ELEMENTS  OF  ALGEBRA. 

Find  the  values  of  aj  in  the  following  equations  : 

4.  x^  +  4:X  =  12.  16.  cc^-7x  =  -6. 

5.  aj2-2a^  =  15.  17.  x'-{-10x  =  56. 

6.  0^  +  80^  =  33.  18.  0)2  +  1205  =  13. 

7.  0)2  + 6a;  =  40.  19.  a;--30x  =  64. 

8.  .T--6o;  =  7.  20.  a;- -f  14 a;  =  32. 

9.  a:2_|.  10^.^11.  21.  3 a;^  +  6 a;  =  504. 

10.  x2_8^.  =  20.  22.  ar-9a;  =  -8. 

11.  a;2  +  20x  =  125.  23.  a;^- 28a;  =  60. 

12.  3a;2  +  6a;  =  24.  24.  a;2^iii^.  =  80. 

13.  a;- 4- 5a;  =14.  25.  a;^  —  15a;  =  —  36. 

14.  2a;2_|_5^.^j[8  26.  a;2  +  2a;  =  99. 

15.  3a;2  — 25a;  =  -50.  27.  a;- -3a;  =  4. 

170.    Other  methods  of  completing  the  square. 

In  the  previous  examples,  when  the  square  of  the  unknown 
quantity  had  a  coefficient,  the  equation  was  divided  by  that 
coefficient  so  that  the  te^^m  containing  the  square  of  the 
unknown  quantity  might  be  a  perfect  square. 

The  coefficient  of  the  second  power  may  always  be  made 
a  square,  all  fractions  avoided,  the  square  completed,  and 
the  values  of  the  unknown  quantity  found  as  follows : 

KuLE.  Multiply  the  equation  by  four  times  the  coefficient  of 
the  highest  power  of  the  unknown  quantity ;  add  to  that  the 
square  of  the  coefficient  of  the  first  power  of  the  unknown 
quantity,  and  then  find  the  value  of  the  unknown  quantity 
by  extracting  the  square  root,  etc. 


QUADRATIC  EQUATIONS.  166 

1 .  Solve  the  equation  3x^  -\-5x  =  S. 

Solution.  3  x^  +  5  x  =  8 

Multiplying  by  12,  etc.,  36  x2  +  60  x  +  25  =  96  +  25  or  121 

6x  +  5:=±  11 

6x=:6  or  -  16 
X  =  1  or  —  I 

2.  Solve  the  equation  ax^  -\-bx  =  G. 

Solution.  '^  ax^  -^  bx=  c 

Multiplying  by  4  a,  etc. ,  4  a^x^  +  4  abx  +  b^  =  4:  ac  -\- b"^ 


2ax-]-b=±  Vi  ac  +  6^ 


2ax  =  —  b  ±  V4  «c  +  6 

x  =  -~  ±  —  V4  ac  +  b 
2a     2a 

Solve  the  following  equations : 

3.  3x'-^x  =  55.  14.   x'-2bx  =  3h\ 

4.  2a;2  +  a;  =  21.  15.    a;- +  3 aaj  =  10 a^. 

5.  4a;2  +  3a;  =  7.  le.    6a^-5a;=76. 

6.  2ar^-9ro  =  -4.  -^^     3a^-x  =  70. 

7.  4a^4-2a;=110. 

8.  2x2-6x'  =  56. 


18.    3x-^  =  26. 


9.   5.'^-4.  =  288.  ^^     20.-^  =  9. 

10.  8ic--6a:  =  2.  ^ 

11.  7a^-4a:=20.  ^0.    x{2x-A)  =  x^ +  12. 

12.  aj2^17aj-18  =  0.  21.   aa;2_4^^^^4 

13.  2a;2-18x  =  -40.  22.    2ax^ -bx  =  2a +  b. 

23.  a;(4a;+3)  =  a.-2  +  90. 

24.  (a;-4)(aj-h6)  =  4aj  +  66. 


156  ELEMENTS  OF   ALGEBRA. 

25.  2x^-5cx  =  -2c\ 

26.  x^-2ax  =  m'-a'. 

27.  {x-\-ay^  =  5ax  —  {x-'ay. 

PROBLEMS. 

171.  1.  Find  two  numbers  whose  sum  is  10  and  whose 
product  is  24. 

Solution.    Let  x  =  one  number. 

Then  10  -  x  =  the  other. 

10  X  -  x^  =  24 
x2  -  10  X  =  -  24 
x2  -  10  X  +  25  rr  1 
x-5=± 1 

x  =  6  or  4 
10  -  X  :zr  4  or  6 

2.  Divide  13  into  two  such  parts  that  their  product  may 
be  36. 

3.  The  difference  between  two  numbers  is  6,  and  their 
product  is  160.     What  are  the  numbers  ? 

4.  The  difference  between  two  numbers  is  4,  and  the 
sum  of  their  squares  is  40.     What  are  the  numbers  ? 

5.  An  orchard  containing  2808  trees  had  2  trees  more 
in  a  row  than  the  number  of  rows.  How  many  rows  were 
there  ?     How  many  trees  were  there  in  a  row  ? 

6.  A  rectangular  piece  of  ground  is  5  rods  longer  than 
it  is  wide  and  contains  500  square  rods.  What  is  the 
length  of  its  sides  ? 

7.  If  four  times  the  square  of  a  certain  number  is 
diminished  by  twice  the  number,  there  is  a  remainder  of 
30.     What  is  the  number  ? 


QUADRATIC    EQUATIONS.  157 

8.  The  difference  between  two  numbers  is  4,  and  the 
sum  of  their  squares  is  346.     What  are  the  numbers  ? 

9.  If  6  times  the  square  of  a  certain  number  is  dimin- 
ished by  10  times  the  number,  the  result  will  be  304.  What 
is  the  number  ? 

10.  ♦  A  sum  of  $  224  is  divided  among  a  certain  number  of 
men  in  such  a  manner  that  each  man  receives  $20  more 
than  the  number  of  men  in  the  company.  How  many  men 
are  there  ? 

11.  There  is  a  certain  number  which,  being  subtracted 
from  20  and  the  remainder  multiplied  by  the  number,  gives 
a  product  of  100.     What  is  the  number  ? 

12.  A  certain  number  of  men  pay  a  bill  of  $240.  If 
each  man  pays  $  1  more  than  the  number  of  men,  what 
amount  does  each  man  pay  ? 

13.  A  lady  paid  $40  for  some  broadcloth.  If  the  price 
per  yard  was  $  6  less  than  the  number  of  yards  she  bought, 
how  many  yards  did  she  buy  ? 

14.  A  man  bought  some  cattle  for  $  100.  If  the  number 
had  been  one  more,  the  price  per  head  would  have  been  $  5 
less.     How  many  cattle  did  he  buy  ? 

15.  A  merchant  sold  goods  for  $  150,  gaining  a  per  cent 
equal  to  one  half  the  number  of  dollars  which  the  goods 
cost  him.     What  was  the  cost  of  the  goods  ? 

Suggestion.     Let  2x=  the  cost;  then,  —  =  the  gain  per  cent, 

and  —  =  the  gain. 
100  ^ 

16.  A  man  sold  a  quantity  of  goods  for  $  39  and  gained 
a  per  cent  equal  to  the  number  of  dollars  which  the  goods 
cost  him.     How  much  did  they  cost  ? 


158  ELEMENTS  OF  ALGEBRA. 


SIMULTANEOUS   QUADRATIC    EQUATIONS. 

172.    The  following  solutions  will  illustrate  some  methods 
of  solving  Simultaneous  Quadratic  Equations. 


r  a?  -|-  ?/  =  5) 
1.    Given  <  >■  to  find  the  values  of  x  and  y. 


Solution. 

x  +  y  =    5 

(1) 

xy=    6 

(2) 

(1)  squared, 

a;2  +  2  5C?/  +  2/2  =  25 

(3) 

(2)  X  4, 

4xy  =  24: 

(4) 

(3) -(4), 

X2  -  2  iC?/  +  ?/2  n:      1 

(5) 

Square  root  of  (5), 

x-y =± 1 

(6) 

(l)  +  (6), 

2  ic  =  6  or  4 

(7) 

ic  =  3  or  2 

(8) 

Substituting  in  (1), 

?/  =  2  or  3 

(9) 

f  X  4-y  = 
2.    Given   -     ^  ;  ^, 
ix'-{-y^  = 

l  to  find  the  values  of  x 
=  52  ) 

and  2/. 

Solution. 

x-]-y=    10 

(1) 

x^  +  y'^=    52 

(2) 

Squaring  (1), 

^2  ^2xy-\-y'^  =  100 

(3) 

(3) -(2), 

2xy=    48 

(4) 

(2) -(4), 

x'^  -  2  xy  -\-  y^  =     4 

(5) 

Square  root  of  (5), 

x-y =±2 

(6) 

W  +  (i), 

2  a:  =  12  or  8 

(7) 

a-  =  ()  or  4 

(8) 

Substituting  in  (1), 

?/  =  4  or  G 

(9) 

(     ^  +2/ 
3.    Given   ]  ^    ^      "^^ 

^     >-  to  find  the  values  of 
■  =  17) 

x  and  2/. 

Solution. 

x  +  y=    5     ^ 

(1) 

2x2+  ?/2  =  l7 

(2) 

From  (1), 

x=    5  —  2/ 

(3) 

QUADRATIC   EQUATIONS.  169 

Hence,  2  x'^  =  50  -  20  y  +  2  ?/2  (4) 

Substituting  (4)  in  (2),      50  -  20  ?/  +  2  2/2  +  ?/2  =  17  (5) 

Collecting  terms,  3  ?/2  -  20  ?/  =  -  33  (6) 

Completing  the  square,  etc. ,  j^  =  3  or  3|  (7) 

Substituting  (7)  in  (1)  a^  =  2  or  IJ  (8) 

{  X  -{-y  =11  ) 
4.    Given  ]    ^       „      ^^  [  to  find  the  values  of  a;  and  ?/. 
t  ar  —  2/^  =  11  ) 

Solution.  x  +  y  =  11  (1) 

x^-y^  =  ll  (2) 

(2)-f-(l)  (Ax.  6),  x-y=    1  (3) 

(l)  +  (3),  2x  =  12  (4) 

x=    Q  (6) 

Substituting  (5)  in  (1),  y=    5  (6) 

Find  the  values  of  the  unknown  quantities  in  the  follow- 
ing equations : 


6. 


?/  =  o  )  (     X'  —     y 

(X  +.V  =    20|  (3x  +    2/  =18| 

(aj^-2/-^  =  120  J  *     (     x'-{-2y'  =  A3^ 

1 4  aj  +  4  ?/  =  5  )  I  a?  +  .V  =  11  > 

'     (         8^77  =  2 )  '     lx'-f-  =  33\ 

..+,=    8.  ..^  +  ,-29. 

,2.  +,=11.  .«^-,-102. 

i2a;2^y2^57;  U^_2/=     3) 

1              a.  +  .V=5|  ^^      ra.  +  2/  =  12| 

lx'-2xy-y'  =  7 )  '     \      xy  =  20\ 


10. 


160 


17. 


18. 


19. 


20. 


21. 


22. 


23. 


^  x-y=    1 

\x'-xy-\-y'' 

(  aj2  _  2^2  ^  20 


ELEMENTS   OF   ALGEBRA. 


f 


.  I  aj  +  2/  =  10 

j  2  a;  H-  3  .?/  =  IG  ) 
12  x"-  2/'  =  46  ^ 
I  a; -2/=    4  I 

(        0^2/  =    90  I 
(a;24-2/^  =  181) 

(Ax''-2y'=    92) 

1  xy  =  lo\ 


{ 


25. 


26. 


27. 


28.     ^ 


aj2  4-a'2;  +  7/'^=13i' 


(    ic-?/  =  9) 


^  +  2/^3 
a      h 


PROBLEMS. 

173.  1.  The  sum  of  two  numbers  is  16,  and  their  product 
is  63.     What  are  the  numbers  ? 

2.  The  sum  of  two  numbers  is  11,  and  the  difference  of 
their  squares  is  55.     What  are  the  numbers  ? 

3.  A  is  4  years  older  than  B,  and  the  sum  of  the  squares 
of  their  ages  is  976.     What  are  their  ages  ? 

4.  The  distance  around  a  rectangular  picture  is  56 
inches,  and  its  square  surface  is  192  square  inches.  Find 
the  length  and  the  breadth  of  the  picture. 

5.  There  are  two  unequal  square  fields  which  together 
require  100  rods  of  fence  to  inclose  them.  If  the  sum  of 
their  areas  is  325  square  rods,  what  is  the  length  of  a  side 
of  each  ? 


QUADRATIC    EQUATIONS.  161 

6.  The  difference  between  two  numbers  is  10,  and  the 
difference  between  their  squares  is  340.  What  are  the 
numbers  ? 

7.  The  sum  of  two  numbers  is  7,  and  the  sum  of  their 
squares  is  29.     What  are  the  numbers  ? 

8.  A  and  B  travel  a  certain  distance  in  3  days.  A 
travels  10  miles  a  day  more  than  B,  and  the  square  of  the 
distance  B  travels  per  day  subtracted  from  the  square  of 
the  distance  A  travels  per  day  is  500  miles.  What  is  the 
entire  distance,  and  how  much  of  it  does  each  travel  ? 

9.  A  merchant  received  $10  for  a  certain  number  of 
yards  of  linen,  and  f  9  for  20  yards  more  of  cotton  at  10  cents 
less  per  yard.  How  many  yards  did  the  merchant  sell  of 
each? 

10.  The  area  of  a  rectangular  field  is  1350  square  rods. 
If  its  length  and  breadth  were  each  lessened  5  rods,  the 
area  would  be  1000  square  rods.  Find  the  length  and  the 
breadth? 

11.  The  sum  of  two  numbers  multiplied  by  their  differ- 
ence equals  36,  and  the  sum  of  the  numbers  is  18.  What 
are  the  two  numbers  ? 

12.  A  merchant  has  200  yards  of  silk  and  velvet.  If  50 
times  the  number  of  yards  of  silk  is  subtracted  from  the 
square  of  the  number  of  yards  of  velvet,  the  remainder  is 
400.     How  many  yards  are  there  of  each  ? 

13.  A  rectangular  and  a  square  field  joined  each  other. 
A  side  of  the  square  field  was  half  the  length  of  the  rectan- 
gular field,  but  less  than  its  width,  and  the  area  of  both  fields 
was  2^  acres.  What  were  the  dimensions  of  each  field,  if  it 
required  90  rods  of  fence  to  inclose  them  as  one  field  ? 

milne's  el.  of  alg. —  11. 


GENERAL   REVIEW. 


I. 

174.    1.   When  a  =  l,  h  =  2,  c  =  4,  d  =  6,  what  is  the 
value  of    2a  +  h'-ah  +  2c_cd-bc^ 
d  —  c-\-  ah  cd  —  b 

2.  Find  the  sum  of  3a{x  —  y),  4:a{x  —  y),  2h{x-'y), 
and  ^{x  —  y). 

3.  Simplify  (aj+ 5)  -  (x  +  10) -[a^  -  (3aj  +  25)  -  10]. 

5.  Add  ax{a-l)  +  {W-2)^y\  2(b'-2)-Sy' 
+  3aa;(a~l),  Si/^- 6a.T((X  - 1)  -  6(62- 2),  ^^^j  subtract 
from  the  result  4  ax  (a  —  1)  +  (&-  —  2)  —  7  2/^. 

6.  What  number  added  to  three  times  itself  equals  ab? 

7.  A  man  had  property  costing  4  6  dollars,  7  c  dollars, 
5  6c  dollars,  which  he  sold  for  f  600.  How  much  did  he 
gain? 

8.  From  205  aj^  __  74  /  +  89  c^  subtract  35  a;^  -  4  2/' -  c- 
+  15. 

9.  Simplify  x'^  -f  y'^ -\x^  -  y^ -{2a^  -  i  f-)  -2z\, 

10.  What  number  subtracted  from  a  times  itself  equals 
a^-l? 

11.  Multiply  a;^  +  2  0^2/  +  4  a^V  _^  g  ^j?/^  ^  16  ?/^  by  a;  -  2  y. 

12.  Expand  (a;+3)  (a;-  4)  {x  -f  4)  (a;  -  3)  {x  -  1)  (a^  +  1). 

162 


GENERAL  REVIEW.  163 

13.  Factor  5  x^y  —  10  axy  -f-  25  xy"  —  15  aVy\ 

14.  Factor  it-- -f  X  —  156,  a;-  — 15  a? +  56,  a;-  — 3  a;  — 70. 

15.  If  7  quarts  of  milk  cost  1  cent  less  than  a  cents, 
what  will  b  quarts  of  milk  cost  ? 

16.  With  a  dollars  a  man  paid  for  10  bushels  of  potatoes 
at  c  dollars  per  bushel,  and  received  2  dollars  in  change. 
Express  this  as  an  equation. 

17.  Divide  x'^  +  x^y^  +  y^  by  x^  -f-  xy  +  y\ 

18.  Factor  a*^  -  ^>^  125  a.-^  -  64  y^,  a;^-81. 

19.  Divide  2  a;^  -  9  x^  -  8  a;^  -  1  by  x^  +  3  a;^  _^  3  .^  _|.  j^^ 

20.  Find  the  value  of  x  in  the  equation  bcx  —  ab  =  —  dx 
-1. 

21.  Factor  4a2  +  36a  +  81,  100x^-25y^,  9x^-A2xy 
+  49/. 

22.  A  merchant  sold  6  pounds  of  coffee  at  b  cents  a 
pound,  5  pounds  at  c  cents  a  pound,  and  b  pounds  at  a  cents 
a  pound.  What  was  the  average  price  received  per  pound 
for  the  coffee  ? 

23.  Find  the  value  of  x  in  the  equation  ax  —  a^  —  b^ 
=  2ab-  bx. 

24.  Factor  8  a;^  +  512,  a'b^  -  1,  m^  +  27  7i^ 

25.  Solve  (a;  -  5)  (x  +  4)  =  a;(a;  -  5)  +  8. 

II. 

26.  Write  out  the  following  squares  : 

(x  +  2yy,    (4.a-3by,    {2m-5)%   (3ab-{-l)%   (5  +  2ay, 
(i^x-ly)%  (a2+10)2,  {m'-4.ny,  (x^-Syf,  {3a  +  5xy. 


164  ELEMENTS  OF   ALGEBRA. 

27.  Write  out  the  following  products : 

(a;  -\-7){x-  4),  (x  +  15)  (x  -  15),  {x  +  10)  {x  +  3), 
{x-12)(x-{-9),  {2x+5){2x-o),  {m' +  1)  (m' ^  1) , 
(aj_8)(.T-ll),   (a; +  15)  (a; -14),   (3  a; +  7)  (3  a;  -  7). 

28.  Find  the  highest  common  divisor  of 

3a5_48a,  2a%-16b,  and  5a'c-20c. 

29.  Find  the  highest  common  divisor  of 

x'-\-6x  +  9,  x^-\-x~6,  Sci^-]-7x-6. 

30.  Find  the  lowest  common  multiple  of 

ar^_j_2a;-3,  ar'-3a;  +  2,  x'  +  x-6. 

31.  Find  the  lowest  common  multiple  of 

x^-9,  a;2  +  9 a;  +  18,  x'-j-Sx-  18. 

32.  Eeduce  — ^^ ± —  to  its  lowest  terms. 

a^  —  b' 

33.  Eeduce  — — ^^-^ — ^^   ,  to  its  lowest  terms. 

Qir  —  3xy  —  10y' 

4  2    ^ . 3x-y 


34.  Simplify /^—i_  +  _?_y 

\x-\-y     x-yj       x-{-y 

o.,     ci-      vi?        5         /2a;2  +  12     x''-{-9\ 

35.  Simplify  -j-^x^—± t-j- 

36.  Simplify  ^ ^ +         ^ 


4(1  +  ^)      4(.v-l)      2(2/2-1) 

37.  Simplify       ^(^-^)       x      ^'^^  +  ^^>      . 

38.  Simplify  t^:^^  ^ax -\-  x\ 

39.  Simplify  3a- 4^>  _2a  -  6  -  c      15a  -  4c^ 

^    ^         7  3  12 


GENERAL  REVIEW.  166 

40.  A  can  do  a  piece  of  work  in  m  days ;  B  can  do  it  in 
7?,  days.  Express  the  part  that  each  can  do  in  one  day. 
Express  the  part  that  both  can  do  in  one  day,  and  the  num- 
ber of  days  in  which  both  can  do  the  work. 

41.  Solve  ^^i+^+ 10  =  '^-^  +  ^^±1. 

7  5  3 

42.  Solve  x{x  -  3)  4-  ^  =  a!(a;  -  5)  -f  — . 


43.    Solve 


x^  —  4,      ar— 4      x-\-2 


44.  Solve  "(^^-^)-^(^ +  '")  =  «;. 

h  a 

45.  A  farmer  has  one  third  as  many  horses  as  cows,  and 
one  half  as  many  cows  as  sheep.  If  there  are  a  animals  in 
all,  how  many  are  there  of  each  kind  ? 

46.  Solve  -^^ ^  =  ^L^. 

X  —  a      X  —  b       X  —  c 

47.  Solve  — ^  +  -  ■*  ^ 


l-5x     2x-l      ^x-1 

48.  The  sum  of  two  numbers  is  a,  and  the  first  divided 
by  m  equals  the  second  divided  by  n.  What  are  the  num- 
bers ? 

49.  The  mth  part  of  a  certain  number  plus  10  equals  m 
times  the  number  minus  5.     What  is  the  number  ? 

50.  If  a  certain  number  is  divided  successively  by  a,  6, 
and  c,  the  sum  of  the  quotients  will  be  10.  W^hat  is  the 
number  ? 


166 


ELEMENTS  OF  ALGEBRA. 


IIL 

Solve  the  following  equations  : 

\l0x-9y  =  2l 


54. 


55. 


51. 


52. 


••^1=    6 


x_±_l 
4 

X  — 


+  2/ =  15 

2      10-a^_7/-10^ 


53.   ^ 


L^    y 


2y  +  4     2a;  +  y_^'  +  13 


r 


8 


56.   < 


4  ic  4-121/ =  5 
6  a;-    3:2=2 
[16a;-    dz  =  l 


C^^'I^   ^  =  62 
2     3       4 


:17 


^  o     ^'^     a^  —  10 

Oo.     -—  — 


4      5      12 

I3     4"^    5 

'  x-\-  y  =  a' 

57.  J  aj+  2;  =  6 

J/+  2;=  c 

50- 

fo:^ 

59. 


5 
2      1-aj 


l  +  i» 


60. 


0?  4-4      a;  —  4 
a;  — 4      a;  +  4 

61.    a^-3aj  =  4. 


15 


10 
3  * 


=  '         25 

62.  x''-{-15x=34:. 

63.  4a;2-a;  =  33. 

64.  a;^  —  2  aa?  =  m^  —  a^. 

65.  2aa^— 5a;=2a-|-5. 

66.  5  X- 4- 4  a;  =  9. 


GENERAL   REVIEW.  167 

Solve  by  factoring : 

67.  a;2-5a;-104  =  0. 

68.  a;2-18a;  +  72  =  0. 

69.  x'--12x-\-30  =  5S, 

70.  What  quantity  multiplied   by  itself  gives  x'^-j-4:xy 

71.  Extract  the  square  root  of  60,025. 

Find  the  values  of  x  and  ?/  in  the  following  equations : 

(20.^+    r  =  59)  Ix  -y  =    2\ 

^3     |a;^  +  r  =  37|  ^^     cx'^2xy-f-  =  73^ 

'   \     Sxy  =18)  '   X  x-y  =5^ 

IV. 

76.  One  half  of  Tom's  money  equals  ^  of  John's,  and 
Tom  has  $12  more  than  John.    How  much  money  has  each  ? 

77.  At  a  certain  election  there  were  two  candidates;  the 
successful  candidate  had  a  majority  of  60,  which  was  2V  ^^ 
all  the  votes  cast.  How  many  votes  did  the  defeated 
candidate  receive  ? 

78.  A  lad  spent  on  July  4th  i  of  his  money  and  6  cents 
more  for  firecrackers,  and  ^  oi  his  money  and  4  cents  more 
for  torpedoes.  If  that  was  all  of  his  money,  how  much 
had  he? 

79.  A  man  bought  a  number  of  sheep  for  $225  ;  10  of 
lem  having  died,  he  sold  f  of  the  re 

f  150.     How  many  sheep  did  he  buy  ? 


168  ELEMENTS   OF  ALGEBRA. 

80.  If  to  the  numerator  of  a  certain  fraction  1  is  added, 
the  value  of  the  fraction  becomes  1 ;  but  if  3  is  subtracted 
from  the  denominator,  the  vakie  becomes  2.  What  is  the 
fraction  ? 

Suggestion.    Let  -  represent  the  fraction. 

y 

81.  A  man  bought  20  bushels  of  wheat  and  15  bushels 
of  corn  for  $  36,  and  15  bushels  of  wheat  and  25  bushels  of 
corn,  at  the  same  rate,  for  $32.50.  What  did  he  pay  per 
bushel  for  each  ? 

82.  The  sides  of  a  rectangular  court  are  to  each  other  as 
3  to  4,  and  their  surface  is  2700  square  feet.  What  are  the 
lengths  of  the  sides? 

83.  If  I  of  the  value  of  a  carriage  is  equal  to  f  of  the 
value  of  a  horse,  and  the  value  of  the  carriage  is  f  20  more 
than  the  value  of  the  horse,  what  is  the  value  of  each  ? 

84.  The  sum  of  two  numbers  is  34,  and  their  product  is 
285.     What  are  the  numbers  ? 

85.  A  merchant,  after  selling  from  a  cask  of  vinegar  15 
gallons  more  than  \  of  the  whole,  found  that  he  had  left 
just  4  times  as  much  as  he  had  sold.  How  many  gallons 
did  the  cask  contain  at  first  ? 

86.  The  difference  between  two  numbers  is  6,  and  one 
half  of  their  product  equals  the  cube  of  the  smaller.  What 
are  the  numbers  ? 

87.  A  company  of  20  men  and  women  paid  a  bill  of  f  75. 
The  men  paid  f  of  the  bill,  and  by  so  doing  each  man  paid 
f  1  more  than  each  woman.  How  many  men  were  there  in 
the  company  ?     How  many  women  were  there  ? 


GENERAL   REVIEW.  169 

88.  A  and  B  own  flocks  of  sheep.  If  A  sells  to  B  10 
sheep,  they  will  each  have  an  equal  number ;  but  if  B  sells 
to  A  10  sheep,  A  will  have  three  times  as  many  as  B. 
How  many  sheep  has  each  ? 

89.  What  are  the  two  numbers  whose  difference  is  8, 
and  whose  sum  multiplied  by  their  difference  is  240  ? 

Suggestion.     Let  x  +  4  and  x  —  4  equal  the  numbers. 

90.  A  grocer  will  sell  1  pound  of  tea,  2  pounds  of  coffee, 
and  1  pound  of  sugar  for  f  1.00 :  or  he  will  sell  2  pounds  of 
tea  and  6  pounds  of  sugar  for  $1.00;  or  he  will  sell  3 
pounds  of  coffee  and  2  pounds  of  sugar  for  f  1.00.  What  is 
the  price  per  pound  of  each  ? 

91.  How  far  may  a  person  ride  in  a  stage,  going  at  the 
rate  of  8  miles  an  hour,  if  he  is  gone  11  hours,  and  walks 
back  at  the  rate  of  3  miles  an  hour  ? 

92.  I  have  two  purses,  one  conta^ining  gold  coins,  and  the 
other  silver  coins.  The  money  in  both  purses  equals  in 
value  twice  the  value  of  the  gold  coins  ;  and  i  the  value  of 
the  silver  coins,  increased  by  i  the  value  of  the  gold  coins, 
equals  f  150.     How  much  does  each  purse  contain  ? 

93.  A  man  bought  a  horse,  a  cow,  and  a  sheep  for  a 
certain  sum.  The  horse  and  the  sheep  cost  5  times  as 
much  as  the  cow,  and  the  sheep  and  the  cow  cost  f  as  much 
as  the  horse.    How  much  did  each  cost,  if  the  cow  cost  f  30  ? 

94.  Find  two  numbers,  such  that  their  product  is  21,  and 
their  product  added  to  the  sum  of  their  squares  is  79. 

95.  A  merchant  sold  some  goods  for  f  96,  thereby  gaining 

as  much  per  cent  as  the  goods  cost.     What  was  the  cost  of 

the  goods  ? 

Suggestion.    Letx  =  the  cost ;  then  —  =  the  gain  per  cent. 

100  ^       ^ 


170  ELEMENTS  OF  ALGEBRA. 

96.  A  and  B  have  the  same  income.  A  saves  ^  of  his, 
but  B,  by  spending  $  100  each  year  more  than  A,  at  the  end 
of  5  years  hnds  himself  $240  in  debt.  What  is  the 
income  of  each  ? 

97.  A  farmer  had  his  sheep  in  3  fields.  |  of  the  num- 
ber in  the  first  field  was  equal  to  f  of  the  number  in  the 
second  field,  and  f  of  the  number  in  the  second  field  was 
J  of  the  number  in  the  third  field.  If  the  entire  number 
was  434,  how  many  were  there  in  each  field  ? 

98.  A  boat's  crew  rows  12  miles  down  a  river  and  back 
again  in  2|^  hours.  If  the  current  of  the  river  runs  2  miles 
per  hour,  determine  their  rate  of  rowing  per  hour  in  still 
water. 

99.  If  the  length  and  the  breadth  of  a  rectangle  were 
each  increased  by  1,  the  area  would  be  120 ;  but  if  they 
were  each  diminished  by  1,  the  area  would  be  80.  Find  the 
length  and  the  breadth. 

100.  A  grass  plot  9  yards  long  and  6  yards  broad  has  a 
path  around  it.  If  the  area  of  the  path  equals  the  area  of 
the  plot,  what  is  the  width  of  the  path  ? 

101.  A  and  B  traveled  toward  each  other  from  two 
towns  247  miles  apart.  A  went  9  miles  per  day,  and  the 
number  of  days  before  they  met  was  3  more  than  the  num- 
ber of  miles  B  went  per  day.     How  far  did  each  travel  ? 

102.  Three  equal  square  lots  have  an  area  of  193  square 
rods  less  than  another  square  lot  whose  sides  are  each  13 
rods  longer  than  the  sides  of  each  of  the  three  equal  square 
lots.     Find  the  length  of  a  side  of  each. 

103.  A  rectangular  plot  of  ground  is  surrounded  by  a 
street  of  uniform  width.  If  the  plot  is  50  rods  long  and  40 
rods  wide,  and  the  area  of  the  street  is  784  square  rods, 
what  is  the  width  of  the  street  ? 


GENERAL  REVIEW.  171 

V. 

104.  Find  the  highest  common  divisor  of  o?  +  a^,  (a  +  xy, 
and  a-  —  x^. 

105.  Find  the  highest  common  divisor  of  9  x-  —  1,  9a;-f-3, 
and  (3x  +  l)'. 

106.  Find   the  highest  common  divisor  of   x^  —  ^x  —  'S, 

a^  +  4 a;  4- 3;  and  x^  —  bx  —  Q. 

107.  Find  the  lowest  common   multiple   of  3{x^  -\-xy)^ 
5(xy  —  y^),  and  6(x-  —  y-). 

108.  Find  the  lowest  common  multiple  of  a^  +  11a;  +  30, 
x"  +  12a;  +  35,  and  a;^  +  13a;  -f-  42. 

109.  Find  the  lowest  common  multiple  of  12xy{x^  —  i/^), 
2  x{x  +  yY,  and  ^y{x  —  y)\ 

110.  Reduce     ^  !>^  ~~  „\  to  its  lowest  terms. 

4:(a^b  —  ab^y 

a^  4-1 

111.  Reduce  — — ^   „      ^ r  to  its  lowest  terms. 

a^  +  3a2  +  3a-f  1 

Change  to  equivalent  fractions  having  their  lowest  com- 
mon denominator : 

,.o     8a;-f2    2a;~l         ,     3a;4-2 
113.   ^:=^,     .--^.  and        --^  +  4 


iB2_4    a.^  +  a;-6  x'-^^x^^ 

114.    -A_,  _ii^,  and   -^ 


a;-f  1/    iv-  — 2/^  ic^  +  2/^ 

,,.3             5  -,      2a; 

115.    J  ?  and 


1  +  a;    4-f-4a;  1  —  ar^ 


172  ELEMENTS   OF   ALGEBRA. 

Simplify  : 


a  —  X      a-\-2x      (a  —  x)(a  -\-2  x) 
117.    ^+  y 2a?  x-y  —  xr 

y       ^-\-y    y(^^-y^) 

Solve  : 

■•■•g         ^  —  <J     ^  —  5       .    1 

4(a;_l)~6(a;-l)      9 


119. 


120. 


121. 


122. 


7 

6a: 

3(1  + 2a;' 

^), 

a?  +  1      .T  —  1 

:^-l 

17 
^5(a?  +  3) 

+1= 

0 

_  21+2a; 
3(a;+3) 

-2. 

3x-l 

4  a;- 

2      1 

2a?-l 

3a!- 

2      6 

2 

6 

1 

2a?-3      3a?  +  2      a;-2 


123.  am  —  6  H —  =  — 

m       6 

-  -  .     3  aa;  —  2  6      ax  —  a      ax      2 

124.  — =: 

3b  2b  b       3 

125.  ^  =  6c  +  cZ  +  -.       . 

X  X 

126.  — +  — +  — =  2aaj  +  6aj  +  ca?. 
ac      ab      be 

128.    ar^  -  a?  +  1      a?^  +  a?  +  1  ^  2a?  +  1. 
a;— 1  a;  +  l  a?^  —  1 


GENERAL   REVIEW. 
VI. 


173 


Solve  : 

129.    {  4.x-\-2y-3z=13 
3x—5y-{-4:Z=  3 


131.    <! 


130. 


^,2/ 


b  a 


x+y-b ^ 
a 


a-{-y—x 


b 


+  ^-f    ^  =  3 


+  2/- 


6 

z 

12' 


J 


132. 


ax  -{-by  =  c^ 
a  b 


b-\-y     a  -h  a; . 


133. 


r 


La 

r 


y-b 
(^x-a)-^b(y  —  b) 


=  —  a6c  J 


134.     { 


a-\-b 


I  ^ZL^=l 
[   Aab 

Find  the  square  root  of  the  following : 


135.  8836. 

136.  5625. 

137.  106929. 

138.  10.4976. 


139.  .00103041. 

140.  .00450241. 

141.  4782969. 

142.  43046721. 


143.  4a.'^-4ar^  + 13x2-6.^  +  9. 

144.  16x'-\- 24:0^ -^S9a^-\-60x +  100, 

145.  x^  +2xy-^y^-{-6xz-{-6yz-\-9z\ 

146.  z'  —  2z -{-1-^-2 mz  —  2 ?7i  -h  ml 

147.  a;'  -  60.-^  +  17 a^'  -  34ar  +  46a''-  -  40a;  -}-  25. 


174 


ELEMENTS   OF   ALGEBRA. 


Find  the  cube  root  of  the  following : 

148.  356400829.  151.  820.025856. 

149.  521660125.  152.  8653.002877. 

150.  102503.232.  153.  1.371330631. 

154.  Sx''-{-4tSaaf-^  60  a'x'  -  80  aV  -  90  aV + 108  a'x -  27  a«. 

155.  27m«-1087/i^-45mH440m3+105m--588m-313. 

156.  a^  -  3a«  +  8a^  -  6a^  -  6a*  +  8a^  -  3a  +  1. 


VII. 


Solve  : 


157. 


158. 


159. 


leo. 


161. 


(         xy=    7 

(a^  +  2/'  =  50 


} 


162. 


^  xy=12^ 

\x'-2y=    5) 

x  —  y=    1 
.x^-\-xy-\-  y' 


I  x-y=    1| 

\x^-\-xy-\-y^  =  19) 

'-y=  ct\ 


xr-y^ 


{ 


i    x--y=    2) 
(  0^2  _  2/2  =3  20  [ 


163. 


164. 


\l-\- 

6 

[i-^-^J 

a      0 
^x     y         J 

> 

\^-xy-[-y^  =  l^\ 


165.  The  head  of  a  fish  is  8  inches  long.  The  tail  is  as 
long  as  the  head  and  ^  of  the  body,  and  the  body  is  as  long 
as  the  head  and  tail.     What  is  the  length  of  the  fish  ? 

166.  A  tree  is  broken  into  three  pieces.  The  part  stand- 
ing is  8  feet  long.  The  top  piece  is  as  long  as  the  part 
standing  and  \  of  the  middle  piece,  and  the  middle  piece  is 
twice  as  long  as  the  other  pieces.     How  high  was  the  tree  ? 


GENERAL  REVIEW.  175 

167.  A  person,  being  asked  the  time  of  day,  replied  that 
it  was  past  noon,  and  that  ^  of  the  time  past  noon  was  equal 
to  \  of  the  time  to  midnight.     What  was  the  time  ? 

168.  A  yacht,  whose  rate  of  sailing  in  still  water  is  12 
miles  an  hour,  sails  down  a  river  whose  current  is  4  miles 
an  hour.     How  far  may  it  go,  if  it  is  to  be  gone  15  hours  ? 

169.  Three  men  engage  to  husk  a  field  of  corn.  The 
first  can  do  it  in  10  days,  the  second  in  12,  and  the  third  in 
15  days.     In  what  time  can  they  do  it  together  ? 

170.  Fifteen  persons  agree  to  purchase  a  tract  of  land, 
but  three  of  the  company  withdrawing,  the  investment  of 
each  of  the  others  is  increased  $  150.  What  is  the  cost  of 
the  land  ? 

171.  C  and  D  have  the  same  income.  C  saves  -^j  of  his, 
but  D,  by  spending  f  65  more  each  year  than  C,  at  the  end 
of  6  years  finds  himself  $60  in  debt.  How  much  does 
each  spend  yearly  ? 

172.  I  sold  a  bureau  to  A  for  \  more  than  it  cost  me. 
He  sold  it  for  $  6,  which  was  |  less  than  it  cost  him.  What 
did  the  bureau  cost  me  ? 

173.  A  man  agreed  to  work  16  days  for  $24  and  his 
board,  but  he  was  to  pay  $  1  a  day  for  his  board  every  day 
he  was  idle.  If  he  received  $  14,  how  many  days  did  he 
work? 

174.  A  man  can  saw  2  cords  of  wood  per  day,  or  he  can 
split  3  cords  of  wood  after  it  is  sawed.  How  much  must 
he  saw  that  he  may  be  occupied  the  rest  of  the  day  in  split- 
ting it  ? 

175.  A  carriage  maker  sold  2  carriages  for  $300  each. 
Did  he  gain  or  lose  by  the  sale,  if  on  one  he  gained  25  per 
cent,  and  on  the  other  he  lost  25  per  cent  ? 


176  ELEMENTS  OF    ALGEBRA.  ^ 

176.  A  teacher  agreed  to  teach  9  months  for  $  562^  and  ■ 
his  board.  At  the  end  of  the  term,  on  account  of  2  months'  ■ 
absence,  he  received  only  $  409^.  What  was  his  board  ; 
worth  per  month  ?  1 

177.  How  many  acres  are  there  in  a  square  tract  of  land  ] 
containing  as  many  acres  as  there  are  boards  in  the  fence  ] 
inclosing  it,  if  the  boards  are  11  feet  long,  and  the  fence  is  \ 
4  boards  high  ?  : 

178.  The  area  of  a  square  figure  will  be  doubled  if  its  ;j 
length  is  increased  6  inches  and  its  breadth  4  inches.  Find  j 
the  length  of  the  side  of  the  square.  ] 

179.  A  certain  iron  bar  weighs  36  pounds.  If  the  bar  ; 
had  been  1  foot  longer,  each  foot  would  have  weighed  i  a  i 
pound  less.  Find  the  length  of  the  bar  and  the  weight  per  ; 
foot.                                                                                              -  ■ 

180.  The  fore  wheel  of  a  wagon  makes  6  revolutions  i 
more  than  the  hind  wheel  in  going  120  yards.  But  if  the  | 
circumference  of  each  wheel  were  increased  1  yard,  the  fore  ; 
wheel  would  make  only  4  revolutions  more  than  the  hind  | 
wheel  in  going  the  same  distance.  What  is  the.  circumfer-  ] 
ence  of  each  wheel  ?  ■ 

181.  How  many  quantities  each  equal  to  a^  — 2a  +  l  . 
must  be  added  together  to  produce  5a^  —  6a^+l?  \ 

182.  There  is  a  cistern  into  which  water  is  admitted  by  ; 
three  faucets,  two  of  which  are  of  the  same  size.  When  ; 
they  are  all  open  the  cistern  will  be  filled  in  6  hours,  but  \ 
if  one  of  the  equal  faucets  is  closed  the  other  two  will  \ 
require  8  hours  and  20  minutes  to  fill  it.  In  what  time  ; 
can  each  faucet  fill  the  cistern  ?  ^ 

183.  When  or  -{-  y-  =  2i)  and  ocPy  -f  xy^  =  300,  what  are  the  ! 
values  of  x  and  y  ?  I 


QUESTIONS   FOR   REVIEW. 


175.   How  does  the  algebraic  solution  of  a  problem  differ 
from  the  arithmetical  solution  ? 

What  are  the  letters  used  in  algebra  commonly  called  ? 

Define  unknown  numbers  or  quantities. 

What  letters  are  used  to  represent  unknown  quantities  ? 

What  is  an  equation? 

Give  the  sign  of  equality ;  the  sign  of  deduction. 

What  is  an  algebraic  expression  ?     Illustrate. 

How  do  the  uses  of   signs  in  algebra  differ  from  their 
uses  in  arithmetic  ? 

What  is  a  power  ?     Give  an  illustration. 

What  is  an  exponent  ?     Give  an  illustration. 

How  are  powers  named  ?    What,  also,  is  the  second  power 
called  ?     What,  the  third  ? 

What  is  a  coefficient  ?     When  no  coefficient  is  expressed, 
what  is  the  coefficient  ?    Give  an  illustration. 

How  should  quantities  inclosed  in  parentheses  be  treated  ? 

What  is  an  algebraic  term  ? 

Distinguish  between  positive  and  negative  terms.     Illus- 
trate each. 

Distinguish  between  similar  and  dissimilar  terms.    Illus- 
trate each. 

What  is  a  monomial  ?      A    polynomial  ?      A  binomial  ? 
A  trinomial  ?     Give  an  illustration  of  each. 

Explain  what  is  meant  by  using  the  signs  +  and  —  as 
signs  of  opposition. 

milne's  el.  of  alg.  — 12.  177 


178  ELEMENTS   OF   ALGEBRA. 

When  positive  quantities  are  added,  what  is  the  sign  of 
the  sum  ? 

What  is  the  sign  of  the  sum  when  negative  quantities 
are  added  ? 

How  is  the  sign  of  the  result  determined  when  both 
positive  and  negative  quantities  are  added  ? 

What  kind  of  quantities  can  be  united  by  addition  into 
one  term  ? 

How  may  dissimilar  quantities  be  treated  in  addition  ? 

What  are  the  two  cases  in  addition  ?     Give  the  rule. 

Define  known  numbers  or  quantities.  What  letters  of 
the  alphabet  represent   them  ? 

Instead  of  subtracting  a  positive  quantity,  what  may  be 
done  to  secure  the  same  result  ? 

Instead  of  subtracting  a  negative  quantity,  what  may 
be  done  to  secure  the  same  result  ? 

Give  the  three  principles  in  subtraction. 

Give  the  cases  in  subtraction. 

When  is  it  necessary  in  subtraction  to  inclose  the  coeffi- 
cient of  the  answer  in  parentheses  ? 

Give  the  signs  of  aggregation.      What  do  they  show  ? 

How  may  the  subtrahend  sometimes  be  expressed  ? 

When  the  subtrahend  is  inclosed  in  parentheses  and  pre- 
ceded by  the  sign  minus,  what  must  be  done  when  the 
subtraction  is  performed  ? 

Give  the  two  principles  relating  to  parentheses,  or  other 
signs  of  aggregation. 

How  does  the  sign  plus  before  parentheses  aft'ect  the 
quantity  inclosed  ? 

When  a  quantity  is  changed  from  one  member  of  an  equa- 
tion to  another,  what  change  must  be  made  in  its  sign  ? 

What  are  the  members  of  an  equation?  What  is  the 
first  member  ?     The  second  member  ?     Illustrate. 

What  is  an  axiom  ?     Give  the  six  axioms. 


QUESTIONS   FOR   REVIEW.  179 

What  is  transposition?  Give  the  principle  relating  to 
transposition.     Give  the  rule. 

How  may  the  value  of  the  unknown  quantity  obtained 
by  solving  an  equation  be  verified  ?  Illustrate  the  process 
by  an  appropriate  example. 

When  a  positive  quantity  is  multiplied  by  a  positive  quan- 
tity, what  is  the  sign  of  the  product  ? 

When  a  negative  quantity  is  multiplied  by  a  positive 
quantity,  or  a  positive  quantity  by  a  negative  quantity, 
what  is  the  sign  of  the  product  ? 

When  a  negative  quantity  is  multiplied  by  a  negative 
quantity,  what  is  the  sign  of  the  product  ? 

Name  the  four  ways  of  indicating  multiplication. 

Give  the  three  principles  of  multiplication.  Give  the 
cases  in  multiplication.     Give  the  rules. 

What  is  the  principle  relating  to  the  square  of  the  sum 
of  two  quantities  ?     What  is  the  square  of  a;  -f-  y  ? 

What  is  the  principle  relating  to  the  square  of  the  dif- 
ference of  two  quantities  ?     What  is  the  square  of  x  —  y? 

What  is  the  principle  relating  to  the  product  of  the  sum 
and  difference  of  two  quantities  ?  What  is  the  product  of 
(x  +  y){x-y)? 

What  is  the  principle  relating  to  the  product  of  two 
binomials  which  have  a  common  term  ?  What  is  the 
product  of  (x-\-3){x-{-2)? 

What  are  simultaneous  equations  ?      Illustrate. 

Define  elimination.  Give  the  principle  relating  to  elimi- 
nation by  addition  or  subtraction.     Give  the  rule. 

Illustrate  the  method  of  elimination  by  addition  or  sub- 
traction by  the  solution  of  a  problem. 

What  is  the  sign  of  the  quotient  when  a  positive  quan- 
tity is  divided  by  a  positive  quantity  ? 

What  is  the  sign  of  the  quotient  when  a  negative  quantity 
is  divided  by  a  negative  quantity  ? 


180  ELEMENTS  OF   ALGEBRA. 

What  is  the  sign  of  the  quotient  when  a  positive  quan- 
tity is  divided  by  a  negative  quantity,  or  a  negative  quan- 
tity divided  by  a  positive  quantity  ? 

How  is  the  exponent  of  a  quantity  in  the  quotient  found? 

Name  and  illustrate  the  two  ways  of  indicating  division. 

G-ive  the  principles  relating  to  division.  Give  the  cases 
in  division.  Give  the  rule  under  the  first  case.  Give  the 
rule  under  the  second  case. 

What  is  a  factor  ?  Illustrate  by  giving  a  quantity  and 
its  factors. 

What  is  factoring  ?  Give  the  rule  for  factoring  a  poly- 
nomial all  of  whose  terms  have  a  common  factor. 

How  may  a  polynomial  be  factored  when  only  some  of 
its  terms  have  a  common  factor? 

What  is  meant  by  "  square  root "  ? 

Give  the  rule  for  separating  a  trinomial  into  two  equal 
factors. 

When  can  a  binomial  be  resolved  into  two  binomial 
factors  ?     Give  the  rule  for  so  factoring  a  binomial. 

What  is  a  quadratic  trinomial  ?  Illustrate,  by  solving 
an  appropriate  example,  the  method  of  factoring  a  quadratic 
trinomial. 

What  quantity  will  divide  the  sum  of  two  cubes  ?  How, 
then,  may  the  sum  of  two  cubes  be  factored?  Give  the 
factors  of  or^  -f  if. 

What  quantity  will  divide  the  difference  of  two  cube^  ? 
How,  then,  may  the  difference  of  two  cubes  be  factored? 
Give  the  factors  of  a^  —  y\ 

Describe  the  ambiguous  sign.     What  does  it  indicate  ? 

Explain  the  solution  of  the  four  kinds  of  equations  by 
factoring. 

What  is  a  common  divisor?  Illustrate  the  use  of  the 
common  divisor  by  giving  two  or  more  quantities  and  a 
common  divisor  or  factor  of  them. 


QUESTIONS  FOR  REVIEW.  181 

What  is  the  highest  common  divisor  or  factor  of  several 
quantities?  Give  the  principle  relating  to  the  highest 
common  divisor. 

Solve  an  example  in  which  the  quantities  are  monomials. 
Solve  one  in  which  they  are  polynomials. 

What  is  a  multiple  of  a  quantity  ?  What  is  a  common 
multiple  of  several  quantities  ?  Illustrate.  What  is  the 
lowest  common  multiple  of  several  quantities  ?  Give  the 
principle  relating  to  the  lowest  common  multiple. 

Find  the  lowest  common  multiple  of  two  or  more  quanti- 
ties which  are  monomials.  Find  the  lowest  common  multi- 
ple of  two  or  more  quantities  which  are  polynomials. 

Define  a  fraction ;  an  entire  quantity  ;  a  mixed  quantity. 
Illustrate  each. 

What  is  meant  by  the  sign  of  a  fraction  ?  How  should 
it  be  interpreted  ? 

Give  the  principle  relating  to  reduction  of  fractions. 

When  is  a  fraction  in  its  lowest  terms  ? 

Name  the  five  cases  under  reduction  of  fractions,  and 
illustrate  by  solving  examples. 

Define  and  illustrate  similar  fractions ;  dissimilar  frac- 
tions. 

AVhat  is  meant  by  lowest  common  denominator  ?  Illus- 
trate. Give  the  principles  relating  to  the  lowest  common 
denominator. 

Give  the  rule  for  reducing  dissimilar  to  similar  fractions. 

What  effect  has  it  upon  the  equality  of  the  members  of 
an  equation  to  multiply  both  members  by  the  same  quantity  ? 

How  may  an  equation  containing  fractions  be  changed 
into  an  equation  without  fractions  ? 

What  is  meant  by  clearing  an  equation  of  fractions  ? 
Give  the  principle.     Give  the  rule. 

If  a  fraction  has  the  minus  sign  before  it,  what  must  be 
done  when  the  denominator  is  removed  ? 


182  ELEMENTS  OF   ALGEBRA. 

If  a  fraction  is  multiplied  by  its  denominator,  what  is 
the  effect  ? 

What  kind  of  fractions  can  be  added  or  subtracted  ? 

What  must  be  done  to  dissimilar  fractions  before  they 
can  be  added  or  subtracted  ?     Give  the  principles. 

In  what  two  ways  may  a  fraction  be  multiplied  by  an 
^entire  quantity  ? 

In  what  two  ways  may  a  fraction  be  divided  by  an  entire 
quantity  ? 

How  should  entire  and  mixed  quantities  be  changed  before 
multiplying  ? 

How  may  an  entire  quantity  be  changed  to  ^  fractional 
form  ? 

When  may  cancellation  be  used  in  multiplication  ? 

Solve  an  example  in  multiplication,  making  use  of  factor- 
ing and  cancellation. 

Solve  an  example  in  division,  making  use  of  factoring  and 
cancellation. 

Explain  the  reason  for  inverting  the  terms  of  the  divisor 
in  division  of  fractions. 

How  may  an  unknown  quantity  be  eliminated  from  two 
simultaneous  equations  by  comparison  ?  How  by  substitu- 
tion ?     Give  the  rules.     Illustrate  by  solving  examples. 

Give  the  dehnition  of :  an  equation  ;  members  of  an  equa- 
tion ;  transposition ;  axiom ;  simultaneous  equations ;  elimi- 
nation; clearing  of  fractions. 

What  is  meant  by  the  degree  of  an  equation  ? 

Define  and  illustrate  the  term  simple  equation  ;  quadratic 
equation. 

What  is  the  degree  of  the  equation  x  -\-  a  =  b?  Of  the 
equation  x^  -{-2x  =  4:?     Of  the  equation  x  +  xy  =  6  ? 

Describe  the  method  of  solving  simultaneous  equations 
containing  three  unknown  quantities.  Solve  a  set  of  simul- 
taneous equations  containing  three  unknown  quantities. 


QUESTIONS   FOR   REVIEW.  183 

What  is  a  power  of  a  quantity  ?  How  many  times  is  a 
quantity  used  as  a  factor  in  producing  the  second  power? 
The  third  power  ?     The  ?ith  power  ? 

What  is  involution  ?    Give  the  principles.    Give  the  rule. 

How  is  a  fraction  raised  to  any  power  ? 

What  is  the  sign  of  any  power  of  a  positive  quantity  ? 

What  is  the  sign  of  any  even  power  of  a  negative  quan- 
tity ?     Of  any  odd  power  of  a  negative  quantity  ? 

What  is  a  root  of  a  quantity  ?     How  are  roots  named  ? 

What  is  the  radical  sign  ?  What  does  it  indicate  ?  What 
is  the  index  of  a  root  ?  When  no  index  is  written  in  the 
opening,  what  root  is  indicated  ? 

What  is  evolution  ?  Give  the  principles  relating  to  evolu- 
tion. Illustrate  their  application.  Name  the  cases  of  evo- 
lution.    Give  the  rule  under  each  case. 

How  is  the  root  of  a  fraction  found  ? 

Explain  the  method  of  extracting  the  square  root  of  a 
polynomial  by  solving  an  example. 

Give  the  principles  relating  to  square  root  of  numbers. 
Give  the  rule  and  solve  an  example. 

How  is  the  cube  root  of  a  polynomial  found  ?  Illustrate 
by  solving  an  example.    Give  the  rule. 

Give  the  principles  relating  to  cube  root  of  numbers. 
Give  the  rule  and  solve  an  example. 

Define  a  quadratic  equation;  a  pure  quadratic  equation. 

What  is  the  root  of  an  equation  ?  Give  an  example  of  a 
pure  quadratic  equation  and  solve  it. 

What  is  an  affected  quadratic  equation?  Give  an  ex- 
ample. 

How  many  methods  are  given  for  completing  the  square  ? 
Solve  an  example  by  the  first  method. 

Give  the  rule  under  the  second  method  and  solve  an 
example. 

Give  four  types  of  simultaneous  quadratic  equations. 


ANSWERS. 


Page  8.  —  1.  30,  John  ;  10,  James.  2.  20  bu. ;  40  bu.  3.  $  2000, 
A  ;   $  1000,  B. 

Page9.  — 4.  5.  5.  30  yr.,  father;  10  yr.,  son.  6.  15  and  60. 
7.    ^200,  horse  ;  $100,  carriage.  8.    100  pear;  200  cherry  ;  400 

apple.  9.  80yr.,A;  40yr.,  B.  ;  20yr.,C.  10.  7  and  42.  11.  5. 
12.   3,  Jane  ;  6,  Hannah ;  30,  Mary. 

Page  10.  — 13.  4  cents,  ink;  20  cents,  pens;  72  cents,  paper. 
14.   $6,  hat;  -^  12,  trousers  ;  $42,  overcoat.  15.    $5,  first;  $10, 

second  ;  $  15,  third  ;  $20,  fourth.  16.  20  sheep,  1st ;  40  sheep,  2d  ; 
60  sheep,  3d  ;  80  sheep,  4th.  17.  7,  42,  70.  18.  $1000,  A  ;  $100,  B  ; 
$400,  C  ;  $4800,  D.  19.   $12,  silver  watch  ;  $120,  gold  watch. 

20.    360  votes,  A  ;  90  votes,  B.  21.    $25,  cow;  $50,  wagon  ;  $  100, 

horse. 

Page  11.  —22.   $2000,  A  ;  $6000,  B  ;  $6000,  C.  24.   6  acres, 

potatoes  ;  30  acres,  oats  ;  150  acres,  wheat.  25.  $  1200.  26.  7  hens  ; 
14  ducks;  98  chickens.       27.    9  and  27.       28.    9  gal.;  36  gal. 

Pagel2.  —  29.  700,  smaller  ;  5600,  larger.  30.  7.  31.  5  days. 
32.  11  yr.,  son;  55  yr.,  father.  34.  15.  35.  3.  36.  7  cows  ; 
3  horses.     37.    4  bu.  ;  14  bu. 

Pagel3.  —  38.  24.  39.  18.  40.  18  and  27.  41.  12.  42.  20. 
43.  9  marbles.  44.  12.  45.  80,  1st;  160,  2d  ;  20,  3d.  46.  6  and  30. 
47.    20,  40,  and  60.       48.    $200,  A;  $100,  B;  $600,  C.       49.    126. 

Page  14.  —  50.  24,  Henry;  8,  James.  51.  12^,  May;  4^,  Julia; 
2^,  Hattie.  52.  30.  53.  500,  history  ;  1000,  science,  400,  fiction. 
54.    70.         55.    $15,  A;  $9,  B;  $3,  C.         56.    40. 

Page  15.  —  58.    10  apples.  59.    $  5,  1st ;  $  10,  2d  ;  $  35,  3d. 

60.  30,  1st ;  30,  2d  ;  40,  3d.  61.  $500.  62.  48.  64.  14.  65.  4, 
16,  and  14.        66.    5  and  20. 

Page  16.  —67.  $  300,  1st  son  ;  $  100,  2d  ;  $  200,  3d.  68.  $  125, 
1st ;  $  125,  2d  ;  $62S  3d  ;  $  187 J,  4th.  69.  5^,  Emma  ;  10  ^,  Anna. 
70.    10  mi.  ;  8  mi.     "  71.    60.        73.    $  500,  1st  month. 

Page  17. — 74.  llf^mi.,  stage;  34^  mi.,  steamboat;  277f  mi.,  rail- 
road. 75.  12,  Harry  ;  4,  Edward.  '  76.  $  100.  77.  $  2000,  lot ; 
$  4000,  house  ;  $  1000,  furnishing.  78.  10  trees  in  each  of  the  5  rows  ; 
20  trees  in  each  of  the  8  rows.  79.  $  3  per  day  ;  $  1  per  day  board. 
80.   40  yr.        81.    $3. 

184 


ANSWERS.  185 

Page  18.  —  83.    4  quarter  dollars  ;  8  half  dollars.  84.    $  1.25, 

man  ;  $  1.25,  wife  ;  $  .25  son.  85.  30  rods,  length  ;  15  rods,  breadth. 
86.  30  ft.  left  standing  ;  15  ft.  broken  off.  87.  $400,  1st ;  ^  200,  2d  ; 
$  200,  3d.         88.    $  2000,  1st  yr.  ;  $  8000,  2d  yr.  ;  $  32,000,  3d  yr. 

Page  22.  —  1.    72.      2.   480.      3.    72.     4.    360.      5.    13.      6.    10. 

7.  5.  8.  768.  9.  8.  10.  9.  11.  20.  12.  21.  13.  82.  14.  78. 
15.    25.       16.    16.       17.    45.       18.    8.       19.    4.       20.    18.       21.    586. 

22.  35.        23.    37.         24.    26. 

Page  27. —  1.    18  x.        2.   24  ab.         3.    -  20  mn.        4.    -25xhj. 

5.  Wax.       6.    -28  5c.       8.    10a.       9.   xhf.      10.    -  mx.      11.    5?/. 

12.  6  m.      13.    Sbc.      14.    10  xy.      15.    x^z. 

Page28.  — 16.    Snj^n.  17.    11  a-b.  18.    0.  19.   9(aby\ 

20.  7{a-b).  21.  0.  22.  6(x  +  2/).  23.  10(a-^by\  24.  6cd. 
25.    (xi/)^.      2.    5a -7  6.      3.    8x4- 7?/ -2^.       4.    S xy -\- 6 y'^ -\- S z\ 

Page  29.  —   5.    4  c  +  7  d.  6.    6  n  -  2  m.         7.    4  a  +  6  6  +  c. 

8.  3  x^^  +  10  xz'  -  3  xy.  9.  5  ^^  -  4  ?/.  10.  4  a-6 '  -  5  c2  -  a7. 
11.  2a6-l.  12.  12a-2d.  13.  7  m  +  6  ;i-x+5?/.  14.  6?/  +  4^  +  2?(7. 
15.   -x:hj-2xyK     16.  llx'»  +  ?/".     17.  9n!/AV-3ed     18.  9mH3m«. 

19.  22  ax^  -  28  a?/^  +  9  az^.  20.    8  a^b  +  8  a^^^c  -  20  5V/  +  8. 

21.  -10xH17ax-^-8a--'x  +  9a^.       22.   (24-a-2  5)(c+fZ).       23.  2m. 
Page  30.  — 24.    10  ax  +  11  x.  25.    5(x  +  ?/)+ 9^;- 4. 

I.  (a+ 5  +  8)  barrels.  2.  (7>i  +  fZ+6)  cents.  3.  (2  c-f  2  n)  dollars. 
4.    (a  +  5  +  3)  miles. 

.Pa?-:  31.— 5.  (2c +  400)  dollars.  6.  5  +  1;  5  +  2.  7.  4,1  ?7i  years. 
8.    (c  +  fZ  +  75)  dollars.      9.    7  x  ;  ax.       10.    (5  +  c  +  d  +  2)  pounds. 

II.  (m  +  n  +  _p  +  pr  +  r)  dollars.  12.    (a  +  5  +  d)  dollars. 

13.  (a  +  5  +  c  + 25)  dollars. 

Page34.  —  3.  15  x.  4.  4a5.  5.  5m2.  6.  —4xy.  7.  -2x^yH. 
8.  -bmyi.  9.  4x+4?/.  10.  bab-c.  11.  45-4d.  12.  3x  +  2i/. 
13.  x  +  ^.  14.  4a +  35  + c.  15.  2a- 5.  16.  2 x'^  -  5  ^/-^  -  4  x?/. 
17.    (Jxy-\-2z.  18.    (a2  +  52)-3c-^  +  2(Z.  19.   3a(p  +  g)+5. 

20.  4  a;  -  3  y  +  1.       21.  4  x'  +  ?/  +  5  ^  -  3  v.       22.   5  x  -  3(^  +  2)  +  2. 

23.  -2a-  +  5-^-  3c-^+4d^. 

Page  35.  — 2.   a%.       3.    IQx'^yz.       4:.   y-6z.        5.    3x^+13z/^ 

6.  —  6  5  +  2  c.  7.  6  a5  -  8  c  +  2  d  8.  -  x"^  -  8  xy  +  2  x.  9.  5  ax 
-6y-\-2z\  10.  Sx^  -29x^y^  +  Sxy.  11.  -  2  rt5  +  5  5'^  -  8c4. 
12.  6x-i/  — 6xy"^-3x?/-3.  13.  3x'»+4x'"?y"+4?/".  14.  -4(m-\-n^) 
-4(m2  +  n).  15.  6x^  +  (yy^  -  Sz^  +  ivK  16.  -3(p  +  g) 
-8(r+s)-10. 

Page  36.  — 17.  20  mhix^-  +  12  a25c2  -  4  abd.  18.  -  a^  +  c^  -  6  5'^. 
19.  -Sxy.  20.  x^  +  2xV-2?/*.  21.  Ox^y^^  -  11  z/^^.  22.  -  x*" 
—  6  x"^'"  +  5  2/"*.  23.    5  5x^  +  3  ay'^  +  9  —  2  cy.  24.    4  m'^  +  mn 

-9n2-22.  25.  -  2  a-'x  -  x^^?/ -  3.  26.  llxV  +  4?/2  -  ^2  _  5, 
27.  ic_?/_0_i<.  28.  5  a252- 17  5c  +  952-4.  30.  (a  -  2)x 
+  (5  +  2)y.        31.  (2  c-  5)x-  (a  +  c)?/  +  (a  +  3)^. 

Page  37.  —  32.  (c  -  6)x  -  2  a5^  +  (4  a^  -  3)52.  33.  (2  a  -  3  c) 
(ic_i^)  +  (4a5_2a)x.    34.  (a-5)x+(5-c)?/+(c-(Z)^.     35.  (3-5c)x 


186 


ELEMENTS   OF   ALGEBRA. 


-^('  +  a)y-\'iS-a)z.  36. 

37.    -  bx  +  cp.  38.   (7 

39.   (m—  mn  +  n)x^  -i-(m  —  n  +  4)1/ 
(X  4-  ?/)  +  («  +  6)  (X  -  y)  +  (c  +  d). 


(6a- c)i/ +  (2c -^  a)z-}- (6 -\- d)x. 
'2a)x'  +  (5  a  -  8)?/-^  +  (H  +  ab)z\ 
)if'^+(l  -  a)z\  40.    (a-  h) 

)x.      41.  m  —  n.      42.  X  —  1,  X  4-  L 


43.   (125  -  a)  dollars.  44. 

46.    X  =  greater,   x  —  b  =  less. 
48.   (a  —  m  —  n  —  100)  dollars. 
Page  38.  —  49.   (50  -  25  -  a 

51.   (5  +  d  +  c  -/)  dollars. 


(0  6 


41.  m  —  n. 
25)  cents. 
47.     (d 


(a-h) 
X  —  1,  X  4-  1. 
45.    80  -  X. 
/)  —  c)  dollars. 


■  c)  dollars.     50.   (8  a  -  40)  dollars. 


Page  39.  —  2. 

9.    8.         10.    16. 

Page  40.  — 13. 
20.   llm-4  7i.      21 
24.    C-2&  +  8. 
28.  2  X  +  2  ?/  +  2  ^. 
32.    4x.         33.    8x4-45 

Page  44.  —2.  x  =  6. 


3.    11 
11.    4. 
25.     14.    21. 
x'^  — 11  ax. 


4.    4.      5. 


6.   0.      7.    1.      8.    17. 


15. 
22. 


25.    la-hh.  26. 

29.  4^2-  5a  +  ?/  +  1. 
34.    «-6. 

X  =  7.     4.  X  = 


10.     16.   17.     17. 

8a-  +  8?)'  +  2a&. 


Zh~  a-Q. 

30.  8w.  31. 
35.  2a  +  8c. 
=  2.     5.  x  =  8. 


7.     19.    Ux. 
23.  8x?/-5. 
27.    5x. 

2  x'^  4-  2  2/-^. 


7.  X  =  2.     8.  X  =  7.      9.  X  ==  4. 
13.  X  =  5.     14.  X  =  3.     15.  X  =r  2. 
19.    x  =  -l.       20.    x  =  4.        21. 
24.  X  =  6.     25.  X  =  4.     26.  x  =  7. 


10.  X-  8.      11.  x=  5. 
16.  X  =  1.     17.  X  =  2. 
x  =  9.        22.    x  =  2. 
27.  X  =  0.     28.  X  =  5. 


6. 
12. 
18. 
23. 


x  =  3. 
X  =  9. 
X  =  2. 
x  =  6. 


9.  X  =  12. 


30.  x  =  8.  31.  x=:l.'  32.  x  =  -8.  33.  x  =  5.  34.  x  =  4. 
35.  x=15.  36.  x  =  4.  37.  x  =  2.  38.  x  =  8.  39.  x  ==  2. 
40.  x  =  4.  41.  x  =  8.  42.  x  =  15.  43.  x  =  12.  44.  x  =  7. 
45.  X--4.  46.  x  =  5.  47.  x  =  -  1.  48.  x  =  5.  49.  x  =  9. 
50.  X  =  6.  51.  X  =  10.  52.  x  =  ^.         53.  x  =  9.         54.  x  =  8. 

55.    x  =  4.         56.    X  =3  10.        57.    x  =  .3. 

Page  45.— 1.  4.       2.   12.       3.  2.       4.   14.      5.  5.      6.  87  cents, 
1st ;  27  cents,  2d  ;  26  cents,  8d.  7.   8  cents,  youngest ;  7  cents, 

next;  10  cents,  oldest.  8.  10  and  24.  9.  ^2000,  B  ;  f  6000,  A. 
10.    %  12.       11.    5  miles,  down  hill ;  18  miles,  level ;  9  miles,  up  hill. 

Page  46.  — 12.  $85,  poorer  horse  ;  %  115,  better  horse.  13.  100 
gallons,  first  pipe  ;  200  gallons,  2d  pipe ;  800  gallons,  8d  pipe. 
14.  825  cattle.  15.  %  5,  chain  ;  %  55,  watch.  16.  18  women  ;  22 
men;  50  children.  17.  $42,800,  cargo  ;  $77,200,  steamer.  18.  $600, 
19.  8  7500,  house  ;  $  2500  lot. 
4.  -40  a.  5.  -18  a.  6.  -  llOx.  7.  108 x. 
10.  -216x^.  11.  lOx'vV.  12.  45  xV- 
126m%3.  15.  _  125am  16.  4(a  +  ?)). 
6  ax  +  4  a?/  —  10  az.  19.   6  x^y  —  6  xy^z. 

~  x.y^.     21.  a3&  +  2  a-^?>2  _|_  ^53.     22.  -18m% 


clerk;  $1800,  brother. 
Page  50.  — 3.   40. 
8.    -42  6.       9.    51  c^. 
13.    -  64  cW^zK        14. 
17.    -20(x-?/).  18. 

20.  -  12  (f'xHi  +  20  xhf- 


-  21  m'^n'  —  J5  mn^. 
■15()a^6'^4-48a;^6' 


23.  9x^-18x5  +  45x4  +  9x^^-27x2.     24.   108a7>^ 


72  a262. 


25.   2  ax^  —  6  ax-y  —  6  ax2^  +  2  axy'-z^. 
'-h^.         27.    1 2  m.-^n  -  40  m.^n^  -  82  mn"^. 


28.    —  5x^  —  5x^— 5x'^— 5x2— 5x. 
-60a55^c  +  42a^&2. 


29. 


Page  51.-4. 

+  8  a&2  +  6=i.     7 
10.  9a2-49&?. 


^  +  2  xy  +  ?/2. 


5. 

4a2-25  6^.     8.  x2-6x 
11.  6x2 -9x- 6. 


18  a462  _  54  a^b^  +  45  a262c2 

m2  -  n\  6.   a^  +  S  a^-b 

-40.     9.  6?/2  +  18?/^  +  602. 

12.  6a2+rt&-  1562. 


ANSWERS.  187 

13.  6  m^  + 17  mn+ 12  n^.  14.  20)/^-S2yz-\-V2z^.  15.  6  62+ 6c- 40  c-'. 
16.  24  x^  -  148  a:  -  80.  17.  20 m^  -  It)  mn  +25  mi/-  20  ntj.  18.  x^ 
+  7  .T?/  +  10  ij\  19.  1  -  x^.  20.  .r^  -  y^  -2ij  -\.  21.  0  x^  +  8  xy 
-Si/K  22.  2rt^  + a5 -4flc  +  2?>c  -  6^  23.  12.r;-2  +  13.x  ~  14. 
24.  24  rt V?»,  +  48  ac'  +  o  ac'-m  +  0 c ^  25.  1 2  x -?/-  —  48  ?/ '.  26.  x"^  +  2  x// 
+  ,V^  —  x^  —  yz.  27.  «■-  —  2  rtc  +  c-  —ah  +  6c.  28.  2  a-  —  ax  -  at/ 
+  xy  -  x\         29.    2  x-  +  13  x?/  -  0  x  -  30  //  +  15  ?/2. 

Page  52.  —  30.  24 x^z^  +  4 x//-^  -  8 yK  31.  m^  +  2 ijj  -  ;//'  +  1 . 
32.    25x- -  25x- 30.  33.    Dx^  -  33xV' +  28  ?/*.  34.    a  ^iah 

+  6rt'^62  +  4 a6H 6^.      35.    8 x^  -  30 x%  +  54 xy^  -21  y^.     36.    m'  -  n\ 

I.  2.  2.  4.  3.  40  yr.,  father;  10yr.,son.  4.  25 '/,  Samuel  ; 
15^,  John.       5.    ^1500,  A  ;  ,1^500,  B.       6.    10  yr.  hence.       7.    .f  12. 

Page53.  —  8.  2  or  50.  9.  ^xhf-z\  10.  c?/ cents.  11.  {a,b-\-i)a) 
miles.     12.  (4  a  —8)  animals.     13.  (20  a  —  3  c?)  dollars.     14.  12  r?  days. 

15.  2  h  dollars,  A  ;  lU  dollars,  B.  16.  20  m  cents.  17.  (12  6  -  ah) 
dollars.         18.    (600  -  40  c)  gallons.         19.    10  ac  dollars. 

Page  54.  —  1.  x^-\-2xy  -\-  if.    2.  6'^  +  2  5c  +  c^.     3.  w'^  +  4  m  +  4. 

4.  a^  +  2  ax  +  x-.  5.  x"^  +  6  x  +  9.  6.  ?/'^  +  2  ?/  +  1.  7.  4  x-  +  4  xy 
+  ?/-^  8.  m'^  +  4  m;i  +  4  7^■^  9.  0  a-  +  0  a6  +  6-.  10.  4  x'^  +  12  xy 
+  9 ?/2.  11.  a'  +  8 a6  +  16 h'^.  12.  4 jti'^  +  8 7>iH  +  4 ii\  13.  4 x>* 
+  20X+25.  14.  9??i*^  +  6m+l.  15.  4  a2+ 20a6  +  25 /A  16.  a* 
+  2  a-7)2  +  h^.  17.  x^  +  6  X-  +  9.  18.  4  m^  +  12  mhi^  +  9  «*. 
19.  a  62  +  4a6c  +  4c2.  20.  4  xV"^  +  4  xy^- +  ^^.  21.  y^-\-Sy-z^-^l6z*. 
J32.  x+16xH64.      23.  25aH70  a6  +  49//^.      24.   16a*  +  24a'^//^  +  9  6*. 

Page  55.  —1.  a-  -  2  ax  +  x^.  2.  h-  -  2  he  +  c^.  3.  ?»"^  -  2  ??i;i 
+  w^.  4.  x-  -  4  X  +  4.  5.  ?/-  -  2  ?/^  +  ^2.  6.  a^  -  6  ah  +  9  62. 
7.    5-2  _  4  ^^.  _^  4  (.-2.  8.    4  x2  -  8  x?/  +  4  ?/^.  9.    62  -  10  6  +  25. 

10.   ?/2-2?/  +  l.     11.    a6i-4a6+4.     12.   x2-8x+16.      13.    4a2 

-  12  a6  +  9  62.        14.    m^  -  4  mn  +  4  ?^2.         15.    4  62  -  16  hd  +  16  (^2. 

16.  a^  -  4  a262  +  4  6^.  17.  6-c2  -  2  hcxy  +  X-Y2.  18.  4  x*  -  20  x:^y'^ 
+  25  y».        19.   4  a-  -  4  ac  +  cA       20.    9  ??i*  -  6  m2  +  1.       21.    9  m2n2 

-  24  mn  +16.  22.  ?/*  -  12  ?/2  +  36.  23.  16  x*  -  40  x^'i/^  +  25  yK 
24.   a262-4a6c2  +  4c^. 

Page  56.  —  1.    a2  -  6".      2.    m^  -  n^.      3.   a-  -  x2.      4.    4  a2  -  62. 

5.  4x2-?/-^.  6.  a2-16.  7.  4  w2  -  9  ?22.  8.  2/2-1.  9.  x2  -  2.'S. 
10.    4  -  ifK  11.    a-62  -  9c2.  12.    4.m'^  -4  n^.         13.    6^  -  4  <•'-. 

14.  9x2 -64 2/2.       15.    X2-100.       16.    lY^c' -  e'p.       17.    9x^-4?/*. 
18.    25a2-9x2. 
22     16?/^ -49^2, 

Page  57.  — 1.   x 
4.    x'  —  X  —  42.     5. 
+  100.        8.    x2  +  4 

II.  x2-  12  x  +  35. 
14.    x2-9x  +  20. 

Page  58.  —  15.  x2  +  9  x  -  22.  16.  x2  -  29  x  +  100.  17.  x2  +  20  x 
+  75.  18.  x2  -  9  X  +  18.  19.  x2  +  3  x  -  18.  20.  x2  +  21  x  +  108. 
21.    x2  +  2  X  -  120.  22.    .x2  +  2  xj  -  8  ?/2.  23.   x2  -  8  ax  +  7  a2. 

24.   x2  +  16x?/  +  60?/2. 


19.    a^  -  hK         20.    mhi^  -  16. 

21.    x2  -  3(J. 

23.    9  X'  -  16  yK       24.   4  a"62  -  25 C 

^•-  +  7 X  +  12.      2.    X-  +  4 X  -  5.      3. 

x-  -  5x  +  r. 

x'  +  15  X  +  50.     6.    x2  -  10  X  -  39. 

7.    x2  +  25x 

x-32.        9.    x2+  12X  +  27.       10. 

X'  -  6  X  -  72. 

12.    x2+  lOx-56.               13. 

a:-2  +  7  X  -  8. 

188  ELEMENTS  OF   ALGEBRA. 

Page  60.  —2.  x  =  4:  -,  y  =  2,  3.  x  =  6;  y  =  1.  4.  x  --  5  ,  y  =  4. 
^.  x=l;  y  =  S.  6.  x=^S;  y  =  7.  7.  x  =  5;?/  =  7.  8.  a;=:9; 
?/  =  7.  9.  ic  =  l;^  =  5.  10.  x  =  6;  y  =  1.  11.  x  =^S  ;  y  -2. 
12.  x  =  6;  y  =  13.  IZ.  x  =  S  ;  y  =  91.  U.  x  =  2  ^y  =  2.  15.  x  =  8  ; 
?/  =  1.  16.  ?/  =  12  ;  2  =  2.  17.  ?/  =  4  ;  ^  =  5.  18.  x  =  5  ;  ^  =  8. 
19.  x  =  2;  y=l{.  20.  x=15;2/z=15.  21.  x=4:;y  =  2.  22.  2/  =  3 ; 
2!  =  13.     2Z.   X  =  2  ;  y  =  -  2.     2^.   x  =  S;  y  =  2,     25.    x  =  6  ;  ?/  =  L 

Page  61.  —  1.  X  =  6  ;  2/  =  4.  2.  x  =  8  ;  y  ==  6.  3.1^  100,  A  ; 
$200,  B.  4.  10  yr.,  son  ;  40  yr.,  father.  5.  10  marbles  in  one  ;  15 
in  other.     6.    60  A.  at  $  30  ;  40  A.  at  $  40.      7.    9  oranges  :  4  pears. 

8.  $300),  A;  $2000,  B.  9.  5  and  7.  10.  $  100,  horse  ;$  8o,  cow. 
Page  62.  — 11.  $  I  \,  man  ;  $  1,  wife.  12.  20  and  6.  13.  6  one- 
dollar  bills  ;  12  two-dollar  bills.  14.  $  2,  velvet  ;$  3,  broadcloth. 
15.  3  13,  A  ;  .$  11,  B.  16.  5  cents,  pad  ;  3  cents,  pencil.  17.  $  i, 
men  ;  $  1,  boys.       18.    $  U,  sawing  ;  $  ^,  splitting. 

Page65.  —  3.  2.     4.  8.    5.    -2.     6.    -5?/.    7.  5  a6.    8.    -  3  m?i. 

9.  2  a/jx.     10.    -  3  y'-z.     11.    -  3  x^.     12.    3  c. 

Page6S.  — 13.  5x'.  14.  -  xV^.  15.  2xV'-  16.  -4x.  17.  4x3. 
18.  -DC<1.  IS.  -3X.2-.  2D.  8?7^^  21.  60  xz'.  22.  5x^.  23.  ax 
-2xy.    2i.    3xH-5^-^.     23.    2  a^h' -j- 1  a.     26.    ^'^  -  x^;  +  3.     27.    2  c 

—  Id.  23.  2  x^  -  X  —  3  V.  29.  4  7}i;i  -  3  m  —  2  ?i.  30.  5  bxy  —  3  ax' 
-ley-.  31.  X  -  4  ?/^'^  H- 5  a^^  32.  Ox*^  -  2  x^  +  4x  +  5.  33.  3  a  6c 
+  4c-2?>c\  34.  iJ9m^4- 20m'-28m  -  23.  35.  bxhjz'-z 
-2xz'^-(5x:\/\  36.  x-2xH3x5-4x'.  37.  2  x'^?/H  4  x^^  -  ;j  ?/i -f  5. 
38.  3x  -  2  ?/ +  4ax-^  —  1.  39.  x^-^x^z  —  y^z-^xy'z^  —  yz^  40.  10 xV 
+  7  x?/  — 3  ax^;-  — 4  /)?/^^.  41.  2  ?ix'  —  3  x?/  —  4  m?/'^  +  3  nx.  42.  1  —  6 
(x+  ?/)+  ab  (X  +  ?/)-. 

Page  63.  —  6.   x  +  ?/.      7.    ?>^  +  n.      8.    x"^  +  2  xy  +  2/^.     9.    x  +  3. 

10.  x+  5.      11.   a -12  6. 

Page  69. —12.  x-2.  13.  x;2-7.  14.  a^  +  i.  15.  5x-3. 
+  X  +  3.  17.  4a-3  6.  18.  a— ^/6  +  6-^  19.  x2+2x  +  4.  20. 
21.  x-2.  22.  x2+10x-Hl4.  23.  2x-4?/.  24.  17x-l. 
-2ax  +  4x'.     26.    9 x-  -  (i xy -\- 4: y\     27.   2x3-3x'^  +  2x. 

-Sx-y.       29.    x2-5x  +  G.       30.    1  -  3x -f  2x2  -  x^.       1. 

2.  —  cents.       3.    -  apples.       4.    -•         5.   —  cents. 

X  2  a  5 

15                                                                      d 
Page  70.— 6.    —days.         7.    3  a  animals.  8.    ,  James; 

a  c  +  1 

^^,  Henry.  9.   ^^^±^  dollars.  10.   ^^l+1^  dollars. 

c  4-  1  ft  c 

11.  ^JlA±-2  cents.      1.   o(x+ ?/)+9;s-4.      2.   3^2  _  2  62  _  5^;,. 

o 

3.  2x+2?/  +  2^.  4.  3  x"^  +  5  x?/"  +  4  ?/»*  -  4.  5.  l^(m  — n) 
-1(m  +  ?i)-        6-    7x  + 11x2  +  9x3.        7.    J  a  -  ^  6  +  c.       8.    Sx^y 

-  5  z^y'  +  10  x"»2". 

Page  71.— 9.  x2  -  4  5j2  _  (^  +  2/)2 -|- 8.  10.  -  a'^-x  -  2  m  -  U. 
11.    m^+l.        12.    a2+2(i6  +  62^2ac+2ae  +  2  6c+c2+2  6e  +  2c6-f62. 


16.  x2 

a'-\-y\ 

25.  «2 

28.    x2 

X  +  ?/ 

ANSWERS.  189 

13.  x^  -  32  r.  14.  x^  -76x-  240.  15.  a*  -  2  a'^b'^  +  bK  16.  16  a* 
-80a3  +  500a-625.         17.    2a^-\-Sab-\-b-^.         18.    a*-a34-a2_^^  ^^ 

19.  m  -  c  -V  z.  20.  1  -  3ic  +  2  x;^  -  x\  21.  2x^  +  ioc^ -\.  Sx -h  10. 
22.  x  =  2.  23.  x=:4.  24.  x  =  5.  25.  a;  =7.  26.  x  =  10. 
27.  X  =  3.  2S.  x  =  9;  y  =2.  2d.  x  =  2;  y  =  4.  30.  x  =  8  ;  w  =  3. 
31.   X  =  4  ;  2/  =  4.  '  ^ 

Page  72.  —  32.  $250.  33.  40  mi.,  1st  day ;  55  mi.,  2d  ;  60  mi.,  3d. 
34.  7  days.  35.  10  days  at  $1 ;  10  days  at  $1].  36.  ^60,  father- 
$40,  David.  37.  $25,  overcoat;  $30,  clothes;' $5,  boots.  38.  $3. 
39.    33A  pounds  at  5  cents ;  (56,^  pounds  at  8  cents. 

Page  74.-2.  9x(2x-Sy).  3.  ox^y{^y  -^  4x).  4.  I2m7i^ 
(m-4w)-  5.  5ac(6-c  +  36--^).  6.  2x(Sx -\- 2y  -  ix"-).  7.  3A 
(32/ -2 +  4^).         8.    Sa^b(2a^  +  7  -6ab'^).        9.   ah(Ha~c-d). 

10.  6a^b'\a¥  -\-  6  -  7b).  11.  bab'^Sb-^  -^  4ab^  -  ^a'). 
12.     10m2(2?7i  -  5  4-  3m-0.  13.    xy^(l  -^  9 x  +  27  x'p). 

14.  bm{am  -\-  2  n  +  Zn^).  15.  5x''?/^(9?/  +  12 i). 
16.     V6x;'y{;^y^  -  5x  +  7?/-^).               17.    ^a%\4  +  12  a^^^'^  -  w^m. 

18.  5a(5w  -f  15  m-  -  ^n^).  19.    5x3  (3  a  -  72/3  _  n  ^^^ 

20.  15x*(3&-^  +  4?/  -  2z).  21.  36a22/3(2a?/  -  1  -  3?/2). 
2.    (^  +  2/) («  +  &).            3.    (?>  +  3)(a  +  2).  4.    (x  +  3)(2/  +  3). 

5.  (a:  +  2/)(a-6).  6.    (a:?/ +  6)(1  -  ?>).  7.    (?/-!  +  1)(2/ -  1). 

8.  (x2-2/-^)(a;  +  ?/).      9.   (x^  -  &2)(^-2  _^  5),        10.  (rr^  +  2)(2a  -  1). 

11.  (X- ?))(«- 6).  12.  (x2+«)(x+l).  13.  (a-6)(x^4-?/"-^). 
14.    (l-a3)(l +  a-0.       15.    {\-x^){l-x).       16.   x(x  -  y)(y  -  z). 

•17.    «(l  +  2/)(l  +  ^-). 

Page  75. —  1.  (x -\- y)  (x -\- y) .  2.  (x  +  2)(x  +  2).  3.  (2a~b) 
(2  a-  b).  4.    (3 m  +  n) (3 m  +  n).  5.    (x  -  2 ?/) (x-2y). 

6.  (2/ +  !)(?/+ 1).       7.    (a?>-4)(a6-4).       8.    (2  w  -  5)(2  ?i  -  5). 

9.  (2x  +  2?/)(2x  +  2?/).       10.   (x-5)(x-5).       11.   (l  +  ^)(l  +  2:). 

12.  (a^  4-  2  52)  (a2  +  2  &2).  13.  (a;  _  3  0)  (x  -  3  ^).  14.  (x  +  10) 
(x+10).       15.    (m  +  4?i)(''^  +  4w).       16.    (3x  -  3?/)(3x  -  3  2/). 

Page  76.  -  17.    (3  +  a^)  (3  4.  «2).        ig.    (2  a^x  -  b'^)  (2  ^23;  _  52). 

19.  (am-4jO(«m -4w).  20.  (5x2  +  8)(5x2  +  8).  21.  (^2  +  2x2) 
(a2  +  2x-').  '22.  (2bc-Sd){2bc-'Sd).  23.  (a'^ -\- 9)(a^  +  9). 
24.  (lOx- l)(10x- 1).  25.  (mn-{-2)(mn  +  2).  26.  (x«  +  r) 
(X"  +  r)-  27.  (4  +  2  m2)  (4  +  2  m2).  28.  (1-4  x"»)  (1-4  x"»). 
1.  (x  +  2/)(x-?/).  2.  (x2  +  2)(x2-2).  3.  (a  +  3  &)(«  -  3  6). 
4.    (wi  +  l)(m-l).         5.    (ab-\-c)(ab-c).         6.    (x+5)(x-5). 

7.  (llx  + 10z/)(llx-102/).  8.  (4a  +  3?>)(4a-36).  9.  (x^  +  12) 
(x2-12).       10.    (l  +  ^)(l-.e). 

Page  77.  — 11.    (x- +  2/2)(x  +  2/)(a:  -  ?/).  12.    (x  +  4)(x-4). 

13.  (3x  +  6)(3x-6).  14.  (2x  +  5)(2x-5).  15.  (x2  +  9) 
(x  +  3)(x-3).  16.  (m?i  +  8x)(mw-8x).  17.  (2«  +  4?>)(2a-4?>). 
18.    (x2  +  25)(x  +  5)(x  -  5).                        19.    (3x  +  9)(3x  -  9). 

20.  (5 a6  +  7 c^d^) {^ab-7 d^d'-).  21.  {m'  +  25) (?^  +  5) (m  -  5) . 
22.  (15x+10y)(15x-10?/).  23.  (4^/2+ 10)(2?/+4)(2?/-4). 
24.  (ac+3d)(ac-3c?).  25.  (x  +  ll)(x-ll).  26.  (cd+l)(c(^-l). 
27.  (5?}i  +  15w)(5m  -  15?^).  28.  (x^  +  l)(x3  -  1).  29.  (x'"+2/«) 
(x'«  -  2/").       30.    {x^  +  13) (x^'^  -  13). 


190  ELEMENTS   OF   ALGEBRA. 

Page78.— 3.  (a  +  4)(a  +  2).  4.  (.■«  +  7)(x  -  2).  5.  (x  -  7) 
(x+l).  6.  (w-0)(m-l).  7.  (6  +  3)(6  +  2).  8.  (?i  -  2) 
(>i  +  l).  9.  (/i  +  2)(w-l).  10.  (x-9)(x  +  2).  11.  (x  +  ()) 
(x-2).     12.   (a-8)(a-f-3).     13.   (x-h9)(ic-4).     14.   (?/_2^)(2/ -  ^). 

15.  (m-8;0(m  +  2w).  16.  (a  +  8ic)(a +x).  17.  (ic-20)(x+10). 
18.  {a-nx){a  +  bx).  19.  (x-4)(x-10).  20.  (x  +  5?/)(x-4  ?/). 
at.    (m  +  6)0m  +  4).      22.    (a-75)(«  +  36).     23.    (x-8)(x-()). 

21.  (xH-9)(x-2).  25.  (c  +  21)(c  -  20).  26.  (a  +  75)(a  +  46). 
27.  (x«-5)(a^"-4).  28.  (x-*  -  14)(x  +  11).  29.  (?7i  +  17)(m-15). 
30,    (x^  +  7)(x«  +  5). 

Page79.  — 4.    lx-^y)(x'^-xij-\-if).     6.    (a  +  ?>)(a"^  -  «6  +  ft"-^). 

6.  (m  - /i)("*'^  +  »^i«  +  »i'0-  7.  (a  -b)(a^  -^  ab  +  b-).  8.  (jji  +  w) 
(m  ■  -  m;i  +  >i 0 .      9.    (x  -  y)  (x-  -\- xy  +  y'^) .      10.   (s  +  0  (^'^ -  ^t  +  ^■-) . 

11.  (c-t?)(c^  +  C(? +  (/-).  12.  (?/  +  l)(2/-^-2/+l).  13.  (x-1) 
{x:'-\-x  -\-  I).     14.    (a-c(^)(a"'H-rt^(?  +  c'-^t?-).     15.    (m  +  s)(?>i-— 9ri+s-). 

16.  (x  —  2/^) (.x'  +  xyz  4-  ?/%').  17.  (a  +  6c) («'^  —  abc  +  ft'c^). 
18.  {mn-\-cd  (m'hi^  —  C(lmu-\-c-d').  19.  {ax  —  by){a-x^-\-abxy+b'^y'-). 
20.    (x  +  l)(a^^-.x+l).         2L    (^-l)(2/-^  +  2/+l). 

PageSO.  — 3.    x  =  ±3.       4.   x  =  ±^.       5.    x=±7.      6.    x=±2. 

7.  x  =  ±4.       8.   x=di^.       9.    x=±3.       10.    x  =  ±4.       11.   ^  =  ±2. 

12.  ic  =  ±2.      13.  ic=±9.      14.  x  =  ±6.     15.  ic=±5.     16.  x  =  ±10. 

17.  x=±\.       18.  x  =  ±4.       19.    x-±5«-       20.    5c  =  ±2c. 
PageSl.  — 23.    ?/ =  9  or  1.     94.    x  =  13or-o.     25.   x  =  11  or  5. 

26.    0  =  2  or  -14.     27.    x  =  3  or  - 17.      28.    x  -  15  or  5.     29.    y  ■-  13  ' 
or  5.        30.    X  —  1  or  -23.       31.   x  =  - 1  or  -5.       32.    x  =  3  or  -27. 
33.   x  =  5or-7.      34.   ?/ =  10  or  4.      35.    0- =  10  or -40.      36.   x=\ 
or -25.      37.    a:  =  10  or -50.      38.    x==27or-l.      39.    ?/ =  6  or -4. 
40.    X  —  15  or  —65. 

Page82.  — 43.  x  =  -5.  44.  x^-Q.  45.  x  =  -?>.  46.  x  =  9. 
47.  x  =  -8.  48.  x=:-13.  49.  x  =  7.  50.  x:=-15.  51.  x  =  14. 
52.  x  =  -10.  53.  x  =  -12.  54.  x=-]8.  55.  x  =  25.  56.  x  =  -ll. 
57.    xr=50.     58.   x=-22.     59.    x  ==  -  30.     60.    x  =  -40. 

Page  83.  —  63.  x  =  5  or  -  3.  64.  x  =  -  5  or  -  1.  65.  x  =  -  9 
or  —  1.         66.    X  =  4.  67.    X  =:  -  8  or  6.         68.    x  =  -  8  or  -  5. 

69.  x=8or-3.  70  x=:-4or-3.  71.  x  =  2or-ll.  72.  x=-5 
or  -  10.  73.  X  =  2  or  -  1.  74.  x  =  7  or  11.  75.  x  =  10  or  -  12. 
76.  xrn  25or  -  3.  77.  x=4a  or  2  <7.  78.  x  =6  or -9.  79.  x  =  12« 
or  -  8  «.     80.    X  =  -  8  6  or  -  3  h. 

Page  84.  —  83.  x  =  -  1  or  -  9.  84.  x  =  2  or  -  10.  85.  x  =  13 
or  5.  86.  x  =  l  or -5.  87.  x  =  15or5.  88.  x  =  lor-15.  89.  x  =  — 
2  or  -  20.  90.  x  =  4  or  -  8.  91.  ?/  =  2  or  -  18.  92.  x  =  17  or  7. 
93.  X  =  24  or  6.  94.  x  =  5  or  -  45.  95.  x  =  14  or  -  0.  96.  x  =  1 
or  -  13.  97.  X  =  22  or  16.  98.  y=\  or  -  3.  99.  x  =  8  or  -  2. 
100.    x  =  2  or  -  10. 

Page86.  —  3.  7x^0.     4.  ^xyz.   5.  5a6.    6.  W  mH'^z.    7.  20a6x*^. 

8.  Qbmhi.     9.    x  —  1.     10.    x -\- y.     11.    m  +  n. 

Page  87.  —  12.  a  -  h.  '  13.  x  +  1.  14.  x  -  2.  15.  y  -  \. 
16.   \  +  a.     17.  x  +  3.     18.  x  +  3      19.*  x- 4.      20.  x^-1.     21.  x-1. 

22.  a  +  6.     23.  x+7.     24.  x-2^.      25.  x-5.     26.  x-4.     27.  x-*2y. 


ANSWERS.  191 

28.  x  +  7  y.  29.  x  -  y.  30.  a:  -  5.  31.  x  -  2  y.  32.  x  -  3. 
33.  a  +  2x.  34.  />  +  c.  35.  x  -  1.  36.  a  +  3.  37.  a-2y. 
38.    2  X  +  6  y. 

Page  89.— 3.    T2a%'^d^.  4.   30a252a.-2.  5.    70 rr-^ftV^Sw. 

6.   891  a27>i  y2.     7.    105  x32/322.      g.   a;^  -  2  x^  -  4  x  H-  8.     9.    r^  +  x'-^?/ 
-  xy-  -  if.       10.    «'2(x'2  -  ?/2).       11.   a^  -  h^. 

5age  90.  —  12.  .x^  -  11  x^  -  4  x  +  44.  13.  x^  +  9  x'^  4-  26  x  +  24. 
14.  xH2x^-16x-32.  15.  c3-9c-'  +  24c-16.  16.  m^x(d^-b-^). 
17.  x\a^  -  b^).     18.   16  rt^  -  8a3  +  2  a  -  1.     19.  a^-4n^-lla-^  60. 

29.  xy{x'-y^).  21.  a?;xy(x3  -  1).  22.  x^  -  a*.  23.  (i^-6aV) 
4.9ah-^-41A  24.  x^  +  Sa^^  _  36x  -  180.  25.  x*  -  x^  4  x  -  1. 
26.  x^  +  8x-2  -  X  -  3.  27.  aJfm{4-^  +  a^/)  -  ab^  -  b^).  28.  105  ab^ 
(a-  -  foO-  29.  x^  +  X-'  -  Ur  -  24.  30.  x^  -  7x2  +  7x  +  15. 
31.  x^  -  2  x^«/  -  52 x^y^  +  98  xy^  -f  147  ?/i  32.  x^  -  6  x-^  -  19  x  +  84. 
38.  x^^-4x-^-47x  +  210.  34.  x^-\-xSf-xy^-yK  35.  x*^-13x- 12. 
36.   x8  -  1. 

Page  93. -2.    1^-        3.    i^^^.        4    l^J^-        5     '^l^llJ^^-V 
30  24  28  *     45  62c3d^ 

g    Pax  —  15  to        -     12??i  —  9?^2        g         4  x'^y  ^  12afec 


33 X  '  48  6ax  +  4x  '    10  ex -6  c?/" 

,Q     16  axy  —  20  bxy 
12  x?/ 
Page  94.  — 11 

14        ^>l  +  n    ^ 
(m  +  70'"^ 

4.   ^.        5. 

7  acx 

10.     '    ■ 

2x-  3a 
Page95.  — 11.    ^^-        12.    — i—       13.    — —-       14.        ^ 


a  —  ?/-  («  —  Z>                w  +  7i  a  —  b 

15.   '^~^.        16.  ^L_.  17.       1_.        18.    «i:/l  19.    "^  +  ^. 

x+?/                 x+1  x+l                 a  —  7  9>i  —  5 

20.    ^-+^.             21.    ^^-^-  22.    ?^ 23.    ''^^. 

X  —  6                     X  —  1  tn-  +  m/i  +  w-  x  +  7 

24. -^^^^ 25.        ^~^ —                      26.    5^ 

x'2  +  xy  -f  t/'2  X-  -x+1  9(x  -  y) 

27. 28.   ^  ~  ^ 


(2x  +  32/)--2  *      2?/ 

Page96.-2.  1^^±A^.  3.  ^l^'-^^ll.  4.  '^^-±J^. 

2  5  4 

,     20x-6y  g     9x-  1  -     5x  -4  g     40x  +  5y 

10        *  '         7      *  '         3      *  '           6        * 

g    40aj-45;-x  ,q     11  m  -  n  ^^     a  -  ?>  +  ex  ^g     18  x  -  ?/  -  g 

5            '  '          3  *        '            c         '  *             9 


192  elemp:nts  of  algebra. 

13.   ?A±_5j2.  14    ^ a;  —  5 y  -g     3 ax-  +  6 g  —  x 

2  7  '  rtx 

jg     a-^  -  2 qc  +  c2  ^^     l  +  2x  ^g       3x2  ^^     4a5-a 

a  *     1  +  X  '    X '  -  1  *  h       ' 

20.    A«!_.     21.  <^'^  +  2  gc  -  2  c^      gg   x-  +  x  -  8      gg   ?n-^  -  4  n^  +  14 
«  —  X         *  a~  c  *x  +  2  '        m  +  2n      ' 

24    m2  +  2  77171  +  n^  -  5        gg     2rt2_5        g^       (m^ 

m+Ti  +  x  *a  +  x 

Page97.  — 2.   ^a-\-~  3.   2rt6  + 

5.   4ac+^.  6.    «+-•  7.   2x- 

9  a 

9.    a  +  X.  10.   x2  +  X  +  ^  "^  ^ 


m 

-  n 

4a& 
11 

4. 

3x?/2^. 

6x 

8.    2 

ax  -  x3. 

11. 

2a 

-26- 

_    2  62 

X-  1  a  4-  6 

12.    5z/-fa^  +  --  13.   2  6 ^.  14.    2x  +  8+    ^^ 


a  a  +  6  X  —  4 

15.    x2  +  x?/  +  2/2  +  -li^.     16.   2  -f  6  -  ^.     17.    X  +  3  +     ^^  +  ^    . 
X  —  w  h  X'-  —  X  —  1 


Page98.  — 18.   2a +  26.  19.   x  -  1.         20.'  2x  +  6  + 


x-3 


21.    a  +  6  +  -?^.  22.    X-2+     —  23.    x-l+ — ^- 

a  —  6  X  —  2  2x  —  3 

24.   6x-2  +  — 5 —      25.   X2-X+1 —- 

X  —  1  X  +  1 

Page  99. -2.   ^,   ^.            3.  ^,   ^^  4.    ^-^^',  -^^. 

6c     he  nx       7ix                    3  6c2/  3  bcij 

g     3a6?/     4c(?        g    2  6^     3cdx  ^    «^     ^2/  a6x        «  ox^ 

12?/'    12?/          *   x2^2'     grV2  *    dxy^    axy  axy  xyz 

bdz      acy     abx       g     2a^      2yz^  2x  ^ 

x?/;^'    x?/;^'    x?/.^            ax?/^'    ax?/^'  ax?/^ 

p        ,QQ  -Q     6c277^         a26c7i      a-7nn        --     10  gxy^     8  by-z 

aVc'^'       a^l)h'-     a-b'-c'^  '     x'^y^z  '    x^y^z' 

9x  *2     12x     52 X     3xy     8?/         ^g     2  ay  -\- 2  by    3 ax  —  3  6x 

x-V'^*  '     78'     78'      78'    78*  '         6  x?/       '  6xy      ' 

a6xy         14    <^cx  +  <^cy     2  ax  —  2  ay     2cx^  +  2c7/2  5  x  —  5 .?/ 

6  xj/  *        4  ac      '  4  ac      '  4  ac  x^  —  7/2 

7 X  +  7  V      x  +  y  jg    3 a  -  3 6    4a  +  46  ^^    3x2 -(ix^ 

x2  —  y-  '   x2  —  7/2*  *     a2  —  62  '     a2  —  62  '    x(x2  —  4) 

5x2+ lOx      9x2-36  j^g      a  +  6  a2  +  a6  +  62       a-b 


x(x^  -  4)       x(x2  —  4)  a-^  -  6>  a^  -  63  a*^  -  6^ 

4x-4  3x-3  4x  ^    ^         '^ 

4(x2-l)'       4(x2-l)'       4(x2-l)* 


19      4x-4  3x-3  4x  g^    2  a2  -t-  2  6 


ANSWERS.  193 


a^  -  b^         '      a^  —  b^  '   x=^— x"^-4a:4-4'   x^  —  x^  —  ix -{- i 

22.  ^^-  5  a;  ax -a 2^       4a;  +  4 

x3  —  3.T-^  —  13x+ 15'  ic^  —  3a;'^  —  18x+ 15  *   4x(a;-2)' 

a;2-x  2x-4  2^    5-  lOx       8x  +  0a:2       4  -  13a; 


4a;(a;-2)        4a;(x-2)  '    l-4x-'        l-4x-='      l-4a:-^* 

gg     a^  -  ab  +  ab'  -  &?        a^  +  ^^^  3a  JlLzlVL 

a^-b^  '        a^  -  6' '       a^  -  6^*  *   y{x'^  +  2/3)' 

xy  +  y'^       x^  —  x^y  +  x-y- 
y{x^  +  y^y      yix^  +  y'O 

Page  103.— 4.  a;  =  8.  5.  a:  =  15.  6.  a;  =  4.  7.  x  =  16, 
8.  a:  =  12.  9.  a;  =  3.  10.  a:  =  6.  11.  a:  =  12.  12.  a;=14.  13.  a; =9. 
14.   a;  =  30.     15.   x  =  24. 

Page  104.  — 16.    x  =  3.     17.   x  =  10.      18.    x  =  24.  19.    x  =  35. 

20.   a;  =  15.       21.   x  =  10.       22.   a;  =  4.        23.   a*  =  44.  24.    a;  =  5. 

25.    x  =  6.     26.    a;  =  12.      27.    x  =  22i\.      28.   a:  =  4.  29.    x  =  l.^. 

30.   x  =  2.         31.    x  =  8.         32.    x  =  13.         33.    x  =  5.  34.    x  =  h 
35.    x  =  3jV     36.    x  =  41.     37.   x  =  17.     38.   x  =  7. 

Page  105.  —  39.  x  =  1.  40.  x  =  9.  41.  x=  50.  42.  x  =  2^\. 
43.  x  =  40.  44.  x  =  16.  45.  x  =  3.  2.12.  3.^7200.  4.  75  gal- 
lons.    5.   10  yeard.     6.  15  and  20. 

Page  106.  —7.  ^75,  horse  ;  i$50,  wagon.  8.  20  and  56.  9.  220. 
10.  10  and  40.  11.  50  years,  father ;  20  years,  son.  12.  100  sheep. 
13.  10  children,  DO  cents.  14.  »^18,  father;  $12,  oldest  son;  $7, 
youngest  son.    ,15.    $4225.      16.    $  135,  A  ;  $140,  B  ;  $165,  C. 

Page  108. -4.    -^  •  6.   ^^^  +  ^^^. 

ab  Sxy 

7.    2x2 +  3y^      ^        2a2         ^    ^a'^-b'^      ^^    ISa 

6 ax  '  aP-  —  x^        '^a  —  A^b  *    12 c      '"'      x-  —  ?/2 

2a^^  14q  +  26  ^g    6x2-9xy  +  4y2  ^^     12m4-2n 

a^  +  8  a  +  15  *     2  x^  —  3  x?/  +  ?/^  *      m-  —  n^ 

2x2  +  2^       jg^   2/2 -2xy       ^^     ^»^L±1.2/l^      18    2  g^  _  6  (:r  +  43 


12. 
15. 
19. 


27. 


x2-l 

2  r?2  -f  2  62 

«2  _  62 

3  a2  _  qa; 
a2  _  x2  ' 

cdx  —  2bz 


xyz 
Page  109.  —  28. 


1  oa    3X+13 


x2  -  3  X  +  2 


OA    4a6 gj    X-  -  9 X  —  9  gg    3x  — 11 


milne's  el,  of  alg,  — 13. 


194 


ELEMENTS   OF   ALGEBRA. 


83. 


zd«^ 34.^^±2i^dols.     35.  ^dols.  paid  out,  2^  left. 


36.    ^^ilMdols.      37.    55^.     38.    fx.     39.    i|a6.     40. 
2  105  ^  ^^ 


6 


41. 


2.3  X 


42. 


39  a  -  28  5 


46. 


20 

x-\-  y  —  2  xy^ 

r^l   _   y'l 

Page  110. —50. 


63 
47. 


43. 


^1  ax 


44. 


5a  +  76 


36  60 

2a-2c 


\-a 
6x-22 


48. 


49. 


x^-x-2 


51. 


ac 
6a;H4a;-6 
x3-7x-6  ' 


45.   ^-±^. 

y 
x-^  - 1  * 


52. 


2ic 


53. 


2a;2 


54. 


2a2 


55     4  d;^  —  4  g'^x  +  4  g-^g 


X*  — 5x^-1-4 
11a, 


x'2  _  1  x^  —  a-x  a^  —  x^ 

57.    ^^-^J^ dollars.      58.    ^^  +  ^^  miles. 
5  3 

Page  112.-3.    "^-^^  4.    1^^.  5.   ^-^.  6. 

ex  9x2  4  5 

8.    ^Jll.         9.      m-  n         ^Q     11  ^2^2        j^^^ 


-  miles. 


7. 
12. 
16. 
21. 


7m2 
6bx 
36 

x  +  2* 
a;  —  5 

x-f3* 
x  +  6 


13. 


4. 

m  —  n 
3(m  +  w) 

14. 


1 


x-y 


17. 


ic'(a  +  x)  5(x  +  1) 

3  ^a  X-hS  ,n  4 


18. 


5(x  -  7) 


19. 


x-S 


2  6m 
6  ^2?/^ 
10  a25 
a:2  -  2/2 
^g^   q(a-36) 
a^  —  6^ 
6(a2-62) 
a 


20. 


Page  113.— 22. 

26.       (x  +  2).     27.    f- 


23.    a. 


24. 


a 
^.    28. 


29.    -\ 


32.   —pounds. 
6 


a 

3  ^22-4  ^: 

33     3m_,K4n  ^^11^,3^ 
3 


25. 


x-y 

30.    1.     31.    —dollars. 
a 

34.    ^^(^-^)  dollars. 
3 


85.   ^,^A',~'^— miles. 
b    b     b        b 


Page  115. —3.    ^. 

my 

25  x3?/5^      a        1 


4. 


5.   X2;. 


6. 


3a62 


7. 


8fl3 


13. 


18. 


4a^bc  X  ■]- y 

dxy        ,^        2 


14. 


2  x^  3  c?x</ 

10.   -^^.     11.   ^^^  +  ^^-^.     12.  ^ . 

a  +  6  a- +  4x2  y{x-y) 

2 


15 


16.   I.      17.   ^iKA+i). 
'^  6 


m{x  —  1) 
a;2_  9x4-20* 


X  -  2  '     x2  +  x?/  +  2/2 

19.    5  6(x-10)  ^^    a-^x  gl.   ^(^  "  ^) 


3  ax 
23.    a-1. 


3x2 


^±^.     25.    4(x-2/)2. 
x-2/ 


ANSWERS.  195 

Page  116.— 26.    -  ^    ^'^^ -•        27.   ^^.        28.   ^dollars. 

29.   2^.     30.   ii?i^zJi  dollars.     31.   3f  cows.     32.   2(a^U^l. 
a  c  (^  3m 

Page  117.  —  2.  X  =  1  ;  2/  =  6.  3.  x  =  2  ;  ?/  =  10.  4.  ic=7  ;  y=2. 
5.x  ==4  ;?/  =  !.  6.  x=:  12;  ?/  =  15.  7.  x  =  18  ;  y  =  14.  8.  x  =  7  ; 
2/  =  7.     9.    x  =  5;  ?/  =  11. 

PagellS.  — 10.  x  =  7;2/  =  2.  11.  x  =  6  ;  ?/ =  5.  12.  x  =  8; 
2/ =11.      13.    x  =  7;?/  =  0.      14.    x  =  6;  2/  =  15.      15.    x  =  4  ;  2/ =  9. 

16.    X  =  18  ;  y  =  12.         17.   x  =  ^^^  ~  ^^ ;  ?/  =  ^^^^^  •         18.   x  =  10  : 

ac  —ah  b  —  G 

2/ =  15.  19.    x  =  — -;y  =  — -.  23.    x  =  9  ;  ?/ =  8. 

4  —  a"^  4  —  a-2 

21.    X  =  5  ;  2/  =  10. 

Page  119.— 2.  x  =  3;  2/=  7.     3.  x  =  6;2/=l.    4.x  =  5;  2/  =  6. 

5.  X  =  10  ;  2/  =  9.     6.    X  =  2  ;  2/  =  8.     7.   x  =  2  ;  2/  =  17.     8.    x  =  19  ; 
2/  =  13.     9.    X  =  12  ;  2/  =  4.      10.    x  =  3  ;  2/  =  4.     11.    x  =  14  ;  2/  =  10. 

Pagel20.  — 12.    x=16;2/  =  12.      13.   x=15;2/  =  18.      14.   x  =  5; 

2/  =  7.     15.    X  =  25  ;  2/  =  40.      16.    x  =  8  ;  2/  =  16.     17.    x  =  9  ;  2/  =  4. 

18.    x  =  la',  y=ib.  19.   x  =  15  ;  2^  =  20.  20.    x  =  ^^  ~  ^^  ; 

,  ,  dm  —  hn 

_  DC  —  ad 

cm  —  an 

Page  121.  — 1.  x=3.    2.  x=6.    3.  x=4^.    4.  x=3.     5.  x=a+&. 

6.  x  =  -^A+_^.  7.   ,,=^±1.  8.   x  =  ^^t^.         9.   x  =  3. 

a  +  c— 1  c'^  —  2  a  —  6 

10.   x  =  40.       11.    x=12.       12.    x  =  14.       13.   x  =  12.       14.    x  =  2. 
15.   x  =  10.     16.    x  =  4. 

Page  122. -17.   x  =  14.         18.   ^  ^  «^±^.       19.   x  = -^. 

ad  -\-  d  a  —  1 

20.    x  =  l.       21.    x  =  4.      22.   x  =  22.       23.    x  =  -7.      24.   x=l}-f 
25.   x  =  l^.     26.    x  =  8.     27.    x  =  4.     28.    x  =  2. 

Pagel2'3.  —  29.  x  =  4^.  30.  x  =  2.  31.  x  =  3^^.  32.  x  =  3. 
33.    x  =  4.      34.    x  =  -lAV     35.   x  =  -  2.     36.    x  =  3.      37.   x  =  6. 

38.    x  =  l.    39.   x  =  tV-    40.   x  =  3.    41.   x  -  a  +  b.     42.   x  =~ 

^^  ab 

Page  124.  —43.  x  =  5  -  1.  44.  x  =  1.  45.  x  =  |-  46.  x  =  2  ; 
y  =  l.  47.  X  =  5  ;  2/  =  7.  48.  x  =  6  ;  ^  =  -  4.  49.  x  =  15  ;  2/  =  10. 
60.  x=6;  y=16.      51.  x=2  ;  y=2.      52.    x=2  ;  y=2,     53.  a:=^^j 

''2  a'  +  b-^'^      a^  +  6^  to+m' 

y  =  ^l-.     56.x  =  mn;y  =  VL.     57.   ^  =  _^  •  j,  =  ^. 
m  —  ?i  ^  a  +  o  a-v  0 


196  ELEMENTS  OF  ALGEBRA. 

Page  125.— -1.    60  and  40.  2.   A  and  B,  9  days  ;  C,  18  days. 

3.  House,  $  3500  ;  invested,  $2500.  4.  72  apples.  6.  20  yards, 
longer ;  8  yards,  shorter.  6.    50  rods,  length  ;  30  rods,  breadth. 

7.    $  800.        8.    500  soldiers. 

Pagel26.  —  9.   A,    120    acres;    B,    180    acres;    C,    150    acres. 

10.  40  apples.  11.  24  rods,  length  ;  12  rods,  breadth.  12.  24. 
13.    %  400.       14.    30  and  40.       15.    100  eggs.       16.    10  and  14. 

Page  127.  — 17.  A,  $300  ;  B,  $500.  18.  $1000.  19.  10  days. 
20.  $  1000.  21.  720  apples.  22.  60  cents.  23.  Horse,  $  100  , 
wagon,  $  50.       24.   16  acres. 

Page  128.  —  2.   x  =  2,  y  z=l,z  =  S.      3.   x  =  i,  y  =  1,  z  =  6. 

Page  129.  —  4.    x  =  5,  ?/  =  3,  0  =  5.  5.    a:  =  4,  ?/  =  10,  0  =  14. 

e.  x  =  6,y^l,  z  =  S.  T.  x  =  S,y  =  7,  z=]l.  8.  x  =  12,  y  =  U, 
z  =  20.  d.   x=  18,  y=\0,z  =  d.  10.   x  =  6,  y  =  6,  z  =  7. 

11.  ic  =  2,  ?/  =  12,  z  =  b.  12.  ic  =  11,  ?/  =1^,  z  =  4.  13.  x  =  10, 
y  =  4:,  z  =  9.  14.  x  =  4:,y  =  4,z  =  S,  15.  x  =  12,  y  =  16,  z  =  20. 
16.  x  =  S,y  =  10,  z  =  S.  17.  x  =  S,y  =  6,z  =  9.  18.  x  =  20, 
y  =  10,z  =  5.       19.    X  =  S,  y  =  5,  z  =  2. 

Page  130.  —20.  x  =  2,y  =  S,z  =  6.  21.  x  =  6,  y  =  6,  z  =  10. 
22.  x  =  10,y  =  U,z  =  13.  23.  x  =  20,  y  =  25,  z  =  30.  24.  x  =  3, 
?/  =  5,  z  =  7.  25.    30  sheep,  1st ;  40  sheep,  2d  ;    60  sheep,  3d. 

26.  10  cents,  Henry  ;  15  cents,  James  ;  25  cents,  Ralph.  27.  150  bu. 
wheat ;  50  bu.  oats ;  40  bu.  barley. 

Page  132.— 3.  4.r.V.  4.  16^^^^  6.  dx'^z^.  6.  Sa^b^ 
7.   -6^a^bV.     8.  Oc'd*.     9.   -x^^zK      10.  126x^'^y^.     11.  81  a^MrA 

12.  16m%4.  13.  -1024a-5mi^  14.  S2  a'^bm^.  15.  -216 a'm\ 
16.  256x32?/8.  17.  -128a2iftii.  is.  - 1 00000  a V^ft^.  19.  ^18^18^15(^9. 
20.    a^^b^hn^.    21.    ~12S  a^^b^H' e\    22.    -x^^^y'^^z'^^^.     23.   iciV"^^"- 

Page  133.— 24.  64x2^*2:48.     25.  -\2^b^^c'^d}^^.     26.  a^%^^^c^^'. 

27.  625a?642;i2^i6.       28.    -aiV^"^^"       29.   2^a^^b''x''y^.     30.    1^. 

9  ?7l'^x2 

g^    27  xy  32       x8y4;gi2  gg     a^^6^^c5»  g^^    125aSx3y3 

Ma^z'^'  '    16a*mi2ri4*  *    x^^^/^'V  *   216  m^n'''^^* 

gg     a^«^2nc3«^         gg      729ffi8x24y^  g^    o^^^  gg    4096  a^6\ 

jc4«^4n^2«*  •     4096  ml2yi62;'24*  *     ?/32^40  *  *     2401  X^* 

39.    «!:!^^    40.  -  . ^^ 41.  -^^!^!^.     1.  x2  -  2  xy  +  ,2. 

5i2n^9«  10000000  ai^m^i  jtiI^ti^Tx^s 

2.   9a2+ i2a6  +  462.  3.   25 ^2  4.  40 «c  +  16 c2.  4.    x3  +  3x2y 

+  3  xz/2  +  ?/^.        5.   x3  -  3  x-?/  +  3  x?/2  -  y^.  6.    a*  +  4  a^^  4.  6  ^2^2 

+  4  a63  _j_  54,  7.  x* -  4 x^/i  4-  6  x-/i' -  4 xw^ + 7i*.  8.  m2  -  6  wm  +  9  w2. 
9.  x2  +  4x2/  +  42/2.  10.  4^2  _20a&  + 2562.  11.  27  x^  +  54x2?/ 
-f  36  x?/2  4-  8  ?/5.  12.  64  x^  -  1 44  xi-y  +  108  x?/^  _  27  y^.  13.  a^  _^  5  a»& 
+  10  a%'^  +  10  a253  +  5  ^51  ^  ^5.  14.  ^^^  +  3  x2  +  3  x  +  1.  15.  «* 
4-2a2624.54,  16.  ^^2  _|.  ^,2  _|.  ^2  _|_  2  a6  +  2  ac  +  2  6c.  17.  «2  +  62 
+  c2  +  2  a6  -  2  ac  -  2  be.  18.   x2  +  ?/2  +  ^2  _  2  x?/  -  2  x^  -1-  2  yz. 

19.  4x2  +  4?/'  +  ^2  4.  8^2/-4x2;- 4?/2;.  20.  a2»  _^  ^2«  _^  c-n  4.  2 «"6»» 
-f  2  a"c"  +  2  6"c".       21.  a-m  4.  yim  4.  ^4"*  __  2  a'"6"*  —  2  a"»c2"*  4-  2  6"»c2'». 


ANSWERS.  19T 

Pagel36.  —  2.  ±2ab.  3.  2x?/2.  4.  ±6x^y'^,  5.  ~Smn. 
6.±2ab'c^.  7.  -a'^b^'c^  8.  -5x3?/^5.  9.  ^a^i)icdK  10.  ±7m%^ 
11.    ±Sa^¥e.  12.    -2a2:;c32,4.  X3.   xV^*-  14.    -4x»y^z^'\ 

15.   ±10a25c^         16.    -Im^xHK         17.    ixV^^^-         18.    ±  a"6*c2". 
19.    -2m%3a,5^8.       20.    i^o^i.      g^,       5m%^.      33^    2a6V.^       , 
3x2?/3  iSa^b^  Zxhfzf 

Page  137. —23.    -|4^-  24.    ^^.  25.    ± 


4  x^xf'z^ 


a^6V  12m87^lo 


Page  139.— 3.  x  +  2y.  4.  a  -  3.  5.  4x  +  32/.  6.  Zx-2y, 
n.  hm^^.  8.  lOa-1.  9.  9a +  5.  10.  6x  +  8y.  11.  a  +  &  -  c. 
12.  x  +  2/4-4.  13.  x-2?/-2^.  14.  2a-3&  +  2.  15.  2a  +  6-l. 
16.  a^-a-2.  17.  2x2+^+1.  18.  3a-c  +  5.  19.  4?7i2+4m-4. 
20.  5a2_i_3cj  +  3.      21.  m2+3m4-l.     22.  ic-?/+2.    23.   m2+3m-27i. 

24.  2a2_5  +  6.    25.    x^-2x+|.     26.   x  +  ?/ -  ^. 

Page  142.  — 3.  27.  4.  81.  5.  98.  6.  73.  7.  85.  8.  66. 
9.  54.  10.  49.  11.  95.  12.  f)7.  13.  235.  14.  163.  15.  118. 
16.  321.  17.  425.  18.  503.  19.  431.  20.  515.  21.  989.  22.  .064. 
23.  4.38.  24.  24.9.  25.  .0795.  26.  3.55.  27.  4.16.  28.  18.22. 
29.    3.71 

Page  144.— 3.    a-\-b.      4.   x-y,      5.   x  +  2.      6.    3a  +  l. 

Page  145.  —7.  3x  -y.  8.  x2  +  4.  9.  2  m  -  n.  10.  x2  +  x, 
11.   «m  +  26.     12.   2a2-5c.     13.    3  +  2&.     14.    1  +  a.     15.  m2  +  6. 

16.  2x-\-Zy.        17.    4x3  +  12?/2.        18.   x^-^x-a'^.         19.    2x-7. 

20.  7x  +  4?/.     21.  a3  +  3ft5.     22.  2x2  +  6z/2.     23.  a%-^.     24.  l  +  3x. 

25.  m'^-n,  26.  x-^-2x  +  l.  27.  d^~?>aAr2.  28.  x2  -  x  -  1. 
29.  3m2-5m-2.  30.  2x2  +  3x?/-?/\  31.  l-3x  +  4x2. 
32.  1  +  4x44  x^. 

Pagel49.  —  4.  8-3.     5.  5-3.    6.  64.    7.  6.3.     8.  89.    9.  97.    10.   68. 

11.  85.  12.  255.  13.  354.  14.  465.  15.  177.  16.  327.  17.  345. 
18.  409.     19.  906.     20.  .25.     21.  352.    22.  237.     23.  .159.     24.  .055. 

25.  3.04.     26.    50.7.     27.    3.64. 

PagelSl.- 3.    x  =  ±4.    4.   x  =  ±3.     5.   x  =  i  5.     6.   x  =  ±6, 

7.  x  =  ±2.     8.    x  =  ±3.      9.   x  =  ±l.     10.   x  =  ±14.     11.    x  =  ±2. 

12.  X  =  ±  5.     13.  X  =  ±  2.     14.  X  =3  ±  8.     15.   x  =  ±  3.     16.    x  =  ±  9. 

17.  x  =  ±10.  18.  x=±(a-3).  19.  x  =  ±2a.  20.  x  =  ±  «•  1.  2. 
2.    6  and  15.    3.    7.    4.    17  rods.    5.    12  yards. 

Page  152.  —  6.    5.  7.    10    rods,   length ;    6   rods,  breadth. 

8.  26  years. 

Page  154.  —4.  2  or  -6.  5.  5  or  -3.  6.  3  or  -11.  7.  4  or  -10. 
8.  7or-l.  9.  lor -11.  10.  2  or -10.  11.  5  or -25. 
12.  2  or  -4.  13.  2  or  -7.  14.  2  or  -4\.  15.  5  or  3^.  16.  6  or  1. 
17.  4  or -14.    18.  1  or -1.3.    19.  32  or -2.    20.  2  or -16. 

21.  12  or  -14.  22.  8  or  1.  23.  30  or  -2.  24.  5  or  -16.  25.  12  or  3. 

26.  9  or  -11.  27.  4  or  -1. 


198  ELEMENTS  OF  ALGEBRA. 

Page  155. —3.  5or-3i.    4.  3  or -3^-.    5.  lor 
7.  5or-5i.    8.  7  or -4.    9.  8  or -7i.     10.   lor-}.     11.  2  or -If. 
12.   1  or  -"'is.        13.    5  or  4.         14.   3  &  or  -  h.        15.    2  a  or  -  5  a. 

16.  4  or -31.        17.    6  or -42.       18.    13  or  -  4i.       19.    6  or  -  li. 

20.  6  or  -  2.  21.  -  1  or  ^^.  22.  ?-^-±_^  or  -  1.  23.  5  or  -  6. 
24.    10  or  -  8.  ^  ^  ^ 

Page  156.  —25.   2  c  or  -•  26.    a  +  m  or  a  -  m.  27.    2  a 

OT\a.  2.  9  and  4.  3.  10  and  16.  4.  2  and  6.  5.  52  rows  ;  54 
trees  in  a  row.     6.    20  rods  ;  25  rods.     7.    3. 

Page  157.  — 8.    15  and  11.  9.    8.  10.   8  men.  11.    10. 

12.    $16.       13.    10  yards.       14.    4  cattle.       15.    $100.       16.    $30. 

Page  159.  — 5.  x  =  2)  y  =  l.  6.  x  =  13  ;  ?/ =  7.  1.  x  =  \ 
or  J  ;  2/  =  J  or  1.  8.  x  =  db  4  ;  ?/  =  ±  4.  9.  x  =  2  or  5^  ;  ?/  =  7  or  i. 
10.  x  =  ±  4 ;  y  =  1  or  9.  11.  x  =  ^)  y  =  b,  12.  x  =  5  or  O^^  ; 
2/  =  3or-li2^.  13.    x  =  7;?/  =  4.  14.    x  =  5  or  2  ;  2/ =  2  or  5. 

15.    X  =  18| ;  2/  =  Vo\.         16.    x  =  10  or  2  ;  ?/  =  2  or  10. 

Page  160.  — 17.  X  =  5 ;  ?/  =  4.  18.  x  =  6  ;  ?/  =  4.  19.  x  =  5 
or  -  9f ;  2/  =  2  or  llf-.  20.    x  =  5  ;  2/  =  1.  21.   x  =  10  or  9  ; 

2/  =  9  or  10.  22.  x  =  11  or  -  5  ;  2/  =  14  or  -  2.  23.  x  =  2  or  —  If  ; 
2/=5  or  — 9.     24.  x=:l  or  5;  2/=4or2.      25.  x=4  or  3  ;  2/=— 3or— 4. 

26.   X  =  8  or  1 ;  2/  =  -  1  or  -  8.  27.   x  =  ^.±1^ ;  y  =  ^Llilk. 

2i  2 

28.   x  =  — ;2/  =  |-         1.9  and  7.       2.   8  and  3.       3.   24  years,  A  ; 

20  years,  B.     4.  16  in.,  length  ;  12  in.,  breadth.      5.   15  rods  ;  10  rods. 

Page  161.  — 6.  22  and  12.  7.  5  and  2.  8.  150  miles,  whole  dis- 
tance ;  90  miles,  A  ;  60  miles,  B.  9.  40  yards,  linen  ;  60  yards,  cotton. 
10.  45  rods,  length  ;  30  rods,  breadth.  11.  10  and  8.  12.  120  yards 
silk ;  80  yards  velvet.     13.  20  rods,  15  rods,  rectangle  ;  10  rods,  square. 

Page  162.  — 1.    2jV       2.    (7a  +  25  +  5)(x  -  2/).       3.    2x  +  30. 

4.    18x2-16x-7\/x.      5.   10?/— 6ax(a-l)-4(&2_2).      6.    — • 

4 
7.    (600  -  4  5  -  7  c  -  5  ftc)  dollars.         8.    170x2  -  7O2/2  +  90c2  -  15. 
9.   2x2-22/2+2^.        10.    a  +  1.        11.    x^ -Z2y^.        12.   x6-26x* 
+  169x2-  144. 

Page  163.— 13.  5x2/(x-2  a+5  2/2-3a2ic2/).  14.  (x+13)(x-12); 
(x-8)(x-7);  (x-10)(x+7).      15.    ^^  ~  ^  cents.      16.    a- 10c  =  2. 

17.  X2-X2/  +  2/2.  18.  (a-62)(a2  4.^^2  +  54).  (5jc_42,)(25x2 
+20x2/+ 162/-^);  (x2  +  9)(x+ 3)(x  -  3).  19.   2x3  -  6x2  +  3x  -  1. 

20.   -^^^zil-.  21.    (2a  +  9)(2a  +  9);  (lOx  +  52/)(10x  -  5?/)^ 

he  -\-  d 

(3x-7  2/)(3x-72/).  22.   ?-^-±l^+-^ cents.  23.   a  +  &. 

11  +  5 

24.  (2x  +  8)(4x2-16x+64);  (a5  +  l)(a6-l);  (m  +  3)(m2-3m  +  9). 

25.  x  =  7.       26.    x2+4x2/  +  42/2;  16a2_24a5+9  62j  4wi2-20m+25; 


ANSWERS.  199 

da^b'^+Qah  +  l,   25+20a4-4a2;    S6x^—Sixy  +  4dy\  a*-h20a^-hl00 ; 
m*  -  8  m^n^  -\-  Wn^;  x*  -  6  x'^y  +  9y^;  9  a'^  -i- SO  ax -\-  25  x-. 

Page  164.— 27.  x2+3x-28  ;  x2-225;  xH13x+30;  x^-3x-108; 
4x2-25;m^-l;x"-19x+88;x2+x-210;  9x2-49.  28.  a -2.  29.  x+3. 

30.  x^- 7x2 +  6.      31.  x3  4-6x2-9x-54.     32.    ~ 33.    ^~^^. 

a  —  b  X  —  by 

34.    -^—  35.    ^^=^.  36.   ^ 37.   — ^— . 

^-2/  3  2(^2  _i)  ^2_a.2 

38     («-^)(<^-a;)         gg    85a  -206 
X  *  *  84        ' 

Page  165.  -  40.  1 ,  A ;  ^ ,  B  ;  '^^i^,  both  ;  -??^,  days.    41.  x= 10. 
m        n  mn  'm-\-n 

42.    x  =  2.        43.    x  =  3.        44.   x  =  a  -  6.        45.   — ,   horses;    ^, 

10'  '    10' 

cows  ;  ^,  sheep.        46.  x  = ^ 47.   x  =  f .        48.    -^^, 

10  a  +  5_c  m  +  7i 

an  ^g       15  m  -^  lOa&c 


m  +  n  m2  —  1  a6  +  ac  +  6c 

Page  166.  —61.  x  =  i;  y=l     52  x  =  2};  y  =  141.     53.  x  =  7  ; 

2^=10.     54.    x  =  --15--;    2/  =  ,-^-  55.    x  =  2]  ;   y  =  ~A; 

oo  —  2a  ba  —  b 

5?  =  4J.  56.    x  =  24;   2/ =  60;   0  =  120.  57.    x  =  ^-±-^^^=-^ ; 

^^«^^^,^^5^|+c.  53    ^^^,  59.    x  =  ±     1 

2  2  a  —  I 

60.   x  =  ±8.         61.   x  =  4or-l.        62.   x  =  2  or -17.         63.   x  =  3 

or  -2J.  64.   x=a  +  mora-  m.  65.   x  =  2«_±_5  or  -1. 

2a 
66.   x=lor  -If. 

Page  167.  — 67.  x  =  13  or  -  8.  68.  x  =  12  or  6.  69.  x  =  14 
or -2.  70.  x  +  2?/-3.  71.245.  72.  x  =  5  or -2f  ;  y=3  or  6J. 
73.  x  =  ±Q;  y=±l.  74.  x  =  lO;  y  =  S,  75.  x  =  ±7;  y="2 
or -12.  76.    $96,  Tom;  $84,  John.  77.    570  votes. 

78.   60  cents.        79.    60  sheep. 

Pagel68.  —  80.  f.  81.  $1.50,  wheat;  $.40,  com.  82.  45  feet, 
60  feet.  83.  $  180,  carriage  ;  ^  160,  horse.  84.  15  and  19.  85.  262^ 
gallons.     86.    x  =  8  or  4}  ;  ?/  =  2  or  —  1.}.      87.    15  men  ;  5  women. 

Page  169.-88.    50  Wep,  A  ;  30  sheep,  B.  89.    11  and  19. 

90.    $.35,  tea;  $.30,  coffee;  $.05,  sugar.      91.   24  miles.      92.    $150. 
93.    $140,  horse;  $30,  cow;  $10,  sheep.      94.    7  and  3.      95.    $60. 

Page  170.  —96.  $312.  97.  162  sheep,  1st;  144  sheep,  2d  ;  128 
sheep,  3d.  98  10  miles.  99.  11  and  9.  100.  1 J  yards.  101.  117 
mi.,  A  ;  130  mi.,  B.       102.    12  rods.       103.    4  rods. 

Page  171.  —104.  a  +  x.  105.  3  x  +  1.  106.  x  +  1.  107.  30  xy 
(x2  -  2/2).         108.    x3  +  18  x2  +  107  X  +  210.         109.    12  xy(x^  -  y^y. 


200  ELEMENTS  OF  ALGEBRA. 

110       ^(<^  +  ^)  -      111      a^^-a+l  .      112    l^Q^+'^^Q       10a; -5 

4abia-b)  '    a-  +  2a-{-l  '    15(x  -  2)  '    15(x  -  2)' 

9x  +  6   ,         113  x3+3x-^-9x-27  x3+3x^-4x-12 


15(x-2)  '   x*+6x3+5x^-24x-36'   xH6xH5x^-24x-36' 

x^  —  16 ---     x^  —  x^y  +  a;-y^  —  x^'^    x^y  +  xy^ 

a:*  +  6x^  + 5x-2-24x-36"  *  x^  -  2/*  '     x*  -  y*  ' 

xV-yy       115     12  -  12  X       5-5x  8x 

x*-?/*'  *    4(1 -xO'   4(1 -x2)'   4(1 -x2)' 

Page  172.— 116.    — 117.       ^    -       118.    x  =  7. 

(a  —  x)(rt  +  2x)  X  -{-  y 

119.  x=:  4.     120.  x=-2i.     121.  x  =  2.     122.  x==l|i.      123.    x  =  to. 

124.    x  =  l.        125.   x  =  ^^^^=J-         126.   x  =  --.  127.   x  =  oSa. 

hc  +  d  abc 

128.    x  =  |. 

Page  173.  —129.    x  =  3  ;  y  =2;  z  =  1.  130.   x  =  2  ;  y  =  4  ; 

^  =  0.      131.    x  =  «;2/  =  2.     132.   x  =  ^' +  ^'- ^^  y  =  ^' "  ^' +  ^^ 

2a  26 

133.    X  =  a  -  _^-^;  2/  =  — ^.       134.  x  =(«  +  6)2  ;  2^  =(«  -  6)2. 

ac  +  6  «c  4-  & 

135.  94.  136.  75.  137.  327.  138.  3.24.  139.  .0.321.  140.  .0071. 
141.  2187.  142.  6561.  143.  2x2  -  x  +  3.  144.  4^2  ^  3^  4.  10. 
145.    X  +  2/  +  3  ^.       146.    z  +  m  -  1.       147.    x-^  -  3  x2  4-  4  x  -  5. 

Page  174.  — 148.  709.  149.  805.  150.  40.8.  151.  9.30. 
152.20.53.  153.1.111.  154.  2x2  +  4 ax  -  3^2.  155.  3m2  -  4m-7. 
156.  a3  -  a2  -  a  +  1.  157.  x=±7;  y=±l.  158.  x  =  8  or  -3  ; 
y  =  l}jOY-4.       159.   x  =  3or-2;    y  =  2or-S.       160.   x  =  ^-±-^; 

y  =  ^^.       161.   x  =  G',y  =  i.       162.    x  =  I  ;  y  =  l       163.    x  =  f; 

y^\'       164.  x  =  ±4;?/=±3.      165.  04  inches.      166.   90  feet. 

Page  175.  — 167.  4  p.m.  168.  80  miles.  169.  4  days.  170.  %  9000. 
171.  ^070,  D;  $005,  C.  172.  $8.  173.  12  days.  174.  11- cords. 
175.    #40,  lost. 

Page  176.  — 176.  #14.  177.  92,160  acres.  178.  12  inches. 
179.  8  feet;  4V  pounds  weight,  per  foot.  180.  4  yards,  fore  wheel; 
5  yards,  hind.  "  181.  5a*  +  4a'^  +  3a2  +  2a  +  l.  182.  211  hours, 
equal  faucets  ;  13  j^^.  hours,  the  other.      183.  x  =  4  or  3  ;  ?/  =  3  or  4. 


XO    86  1* 


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